This post will address the question in 2 ways:
A. A high-level overview comparing/contrasting some of the common views/disagreements on the Holy Ghost
B. A deep dive on my own faith's use of the terms listed in the OP (this was how I read the intent of the question originally)
A. Notable agreements/disagreements
That the Biblical authors do not always use the same terminology in referring to the Holy Spirit does not indicate that one or more of the authors is wrong; one author has a preference for one term, and another author has a preference for a different term. We see this occur in the titles Biblical authors use when referring to the Father and the Son as well.
Many Christian organizations will adopt a specific set of technical terms or preferred terminology, not in an effort to disagree with the Biblical authors who used different terms, but for sake of clarity, consistency, and standardization of their liturgy, doctrine, and teaching.
In other words, a given church may say, "when we use this term we mean X", not because the Biblical text requires that we use this term and only this term, but because this helps us understand what each other are talking about.
One of the clearer divides in Christian thought is on the personhood of the Holy Ghost. The who see the Holy Ghost as a person understand the concept of "the power of the Holy Ghost" as a being exercising power, which is distinct from the view that the power of the Holy Ghost is God's active force. On this latter view, the power of the Holy Ghost is sometimes likened to electricity (source).
Churches that claim apostolic priesthood authority hold a different understanding of terms such as "the baptism of the Holy Spirit" and
"the reception of the Holy Spirit", believing that like John the Baptist acknowledged centuries ago (Matt. 3:11), not everyone can confer this baptism/gift, but only those with specific authority from God to do so.
Those who accept a priesthood of all believers and/or a textual source of authority are not unanimous regarding the necessity of physical ordinances, but do not believe a chain of priesthood authority via the laying on of hands, going back to the apostles, is needful for the baptism of the Holy Spirit or the reception of the Holy Spirit to occur.
One time or many times
Are the indwelling of the Spirit and the filling of the Holy Spirit one time events or repeated events/processes?
Christian apologist William Lane Craig argues that these (or similar) terms describe the baptism/filling of the Holy Spirit is a single experience in the conversion of a believer (source). I argue here that these experiences with the Holy Ghost can and should be recurring events as part of the process of transformation.
The former view would tend to see the powerful manifestations of the Holy Ghost serving to convince someone of the truth; the latter view would accept the role of the Holy Ghost in conversion (e.g. 1 Cor. 2:4), but that it doesn't stop there. The Holy Ghost does not merely seek to impress, but plays an ongoing role in transformation/sanctification.
I acknowledge, of course, that some of these terms (such as #1, #2, #3 and #10 in the OP) are regularly used with overlap in meaning, such that the differences in beliefs among different groups can easily be exaggerated by their use of different terms to say similar things.
Perhaps a fair way to illustrate the distinction, without causing confusion through specific use of technical terms, is to ask: are conversion & rebirth processes or events? As illustrated by the links above, this is not an area of unanimity in Christian thought.
Gifts & fruit of the Spirit
While not necessarily entirely unanimous, these are areas where there is broader agreement among Christians regarding what they are (though Christians will disagree plenty on why or where they are).
This is aided by specific lists provided by Paul:
- Fruit of the Spirit (see Galatians 5:22-23)
- Gifts of the Spirit (see 1 Corinthians 12)
Paul does not claim that either of these lists is exhaustive, but because the lists were provided, Christians will generally agree that things like love & peace are fruit of the Spirit and things like prophesy & tongues are gifts of the Spirit.
How gifts of the Spirit are employed and the purpose they serve is an area, however, where Christians disagree a great deal (including various forms of continuationism & cessationism).
(note that while Christians broadly agree that the gift of tongues is a gift of the Spirit, its function is also an area of disagreement. The two principle views are that it is communication in an unknown/spiritual language (glossolalia) or that it is the ability to communicate in another human language (xenoloalia) for the sake of sharing the Gospel--see here).
B. Deep dive on use of terminology in my own faith
I'll offer the perspective of a member of The Church of Jesus Christ of Latter-day Saints on the terms noted in the OP.
1. The baptism of the Holy Spirit
Generally used to refer to receiving the Gift of the Holy Ghost. Baptism by water is colloquially referred to as just "baptism", and baptism of the Holy Spirit is generally referred to as "confirmation" or "receiving the Gift of the Holy Ghost".
This is illustrated by 2 Nephi 31:13
ye are willing to take upon you the name of Christ, by baptism—yea, by following your Lord and your Savior down into the water, according to his word, behold, then shall ye receive the Holy Ghost; yea, then cometh the baptism of fire and of the Holy Ghost;
(note that "Holy Ghost" and "Holy Spirit" are considered equivalent terms, but in English one will find "Holy Ghost" used much more frequently than "Holy Spirit" in Latter-day Saint conversation)
2. The reception of the Holy Spirit (or receiving the Holy Spirit)
As a technical term, this refers to the result of the ordinance that follows baptism by water: one receives the Gift of the Holy Ghost.
Latter-day Saints distinguish between the Gift of the Holy Ghost--the privilege to be able to have the constant companionship of the Holy Ghost--and the power of the Holy Ghost (source 1, source 2). The effect of the reception of the Holy Spirit is further described by Alma with the words:
that he may pour out his Spirit more abundantly upon you (Mosiah 18:10)
Indicating that there is a degree of ongoing, spiritual outpouring/support that comes only after participating in the ordinances of baptism & confirmation and making the covenants associated with them.
3. The indwelling of the Holy Spirit
"Indwelling" is not a term regularly used in my faith, but similar terminology is found in:
the Spirit of the Lord doth not dwell in unholy temples (Helaman 4:24)
Unworthiness limits our ability to have the influence of the Holy Ghost in our lives.
34 Ammon said unto him: I am a man; and man in the beginning was
created after the image of God, and I am called by his Holy Spirit to
teach these things unto this people, that they may be brought to a
knowledge of that which is just and true;
35 And a portion of that Spirit dwelleth in me, which giveth me
knowledge, and also power according to my faith and desires which are
in God. (Alma 18:34-35)
Ammon, a missionary, acknowledges that it is the Holy Ghost working in him that has given him the ability to do God's work. In this sense, the indwelling of the Holy Spirit would refer to receiving knowledge & power through the Holy Spirit.
4. The sealing of the Holy Spirit
My faith uses "sealed by the Holy Spirit of Promise" as a technical term (see Doctrine & Covenants 132:7), describing the ratification by the Holy Ghost that covenants entered into have been kept.
To seal is to ratify, to justify, or to approve. Thus an act which is sealed by the Holy Spirit of Promise is one which is ratified by the Holy Ghost (source)
An unworthy candidate for baptism might deceive the elders and get the ordinance performed, but no one can lie to the Holy Ghost and get by undetected. Accordingly, the baptism of an unworthy and unrepentant person would not be sealed by the Spirit; it would not be ratified by the Holy Ghost; the unworthy person would not be justified by the Spirit in his actions. (ibid)
5. The inner witness of the Holy Spirit
Revelation from Spirit to spirit--the means by which God (usually) communicates with mortals and provides them with assurance of what is true. See further discussion under #6.
6. The power of the Holy Spirit
Used distinctively from #2, it describes the revelatory role of the Holy Ghost that is available to all. The most quintessential use of the term in Latter-day Saint culture is in reference to Moroni's promise, quoted in part below:
4 And when ye shall receive these things, I would exhort you that ye
would ask God, the Eternal Father, in the name of Christ, if these
things are not true; and if ye shall ask with a sincere heart, with
real intent, having faith in Christ, he will manifest the truth of it
unto you, by the power of the Holy Ghost.
5 And by the power of the Holy Ghost ye may know the truth of all
7 And ye may know that he is, by the power of the Holy Ghost;
wherefore I would exhort you that ye deny not the power of God; for he
worketh by power, according to the faith of the children of men, the
same today and tomorrow, and forever. (Moroni 10:4,5,7)
This is available before baptism, and is the means emphasized by the church's missionaries by which people can know for themselves that the message being taught is true.
This then is closely related to #5. Through the power of the Holy Ghost (the Holy Ghost has the power to act as a revelator) people can receive the inner witness of the Holy Ghost, a testimony of true principles revealed by God which can neither be given nor taken away by human wisdom (see also 1 Cor. 2:4-5).
A brief discussion of my own experience on this matter is found in this video on my channel.
7. The fruit of the Holy Spirit
The effects that the influence of the Holy Ghost brings about--the classic statement on the matter is found in Galatians 5:22-23--this is how the Holy Ghost leads people to act & feel. This is the result of inviting the Holy Ghost into one's life and following the Holy Ghost's guidance.
8. The gifts of the Holy Spirit
Abilities given by God to aid in His work. This includes the ability to communicate in another language or to prophesy, but also less "showy" gifts, such as the ability to organize, or to teach spiritual matters clearly (see Doctrine & Covenants 46:11-26).
11 For all have not every gift given unto them; for there are many gifts, and to every man is given a gift by the Spirit of God. (ibid)
God endows different people with different abilities--and is willing to grant greater support to those who diligently seek His aid--in order to carry out His work. As Paul pointed out through illustrative metaphor, it wouldn't work very well if everyone had exactly the same gifts (hands, feet, noses). Or, as a friend of mine whom I will respectfully leave anonymous put it:
our collective strengths make our individual weaknesses irrelevant.
9. The leading of the Holy Spirit (or the closely related walking in the Spirit)
Receiving guidance from the Holy Ghost and following it. Nephi demonstrated this in obtaining the Brass Plates--he acted in faith, came up with a plan, used his resources to the best of his abilities, and trusted the Lord to make a way forward:
And I was led by the Spirit, not knowing beforehand the things which I should do. (1 Nephi 4:6)
10. The filling of the Holy Spirit
Not a one time event, but a way to describe an experience of being particularly moved by the Holy Ghost. In my faith, the terminology often used is "the Spirit was very powerful in that meeting". See also #3.
Being filled with the Spirit is frequently associated with receiving knowledge and/or power from God.
11. Signs & Wonders
My faith certainly believes in modern miracles (e.g. see Articles of Faith 6, Moroni 7:36-37). However, my experience is that miracles do not produce faith, faith produces miracles. I have written further on this topic here.
Sometimes I am asked why I do not seek to impress anyone with spiritual experiences I have had. Getting it straight from the source (God) will be a lot more impressive than anything you hear from me. I do not believe that a knowledge of God's existence, plan, and love, is intended to be a second-hand experience.
12. Sanctification by the Spirit
One of the more effective descriptions of the work of the Holy Ghost is given by the Savior in 3 Nephi 27:20
20 Now this is the commandment: Repent, all ye ends of the earth, and come unto me and be baptized in my name, that ye may be sanctified by the reception of the Holy Ghost, that ye may stand spotless before me at the last day.
Fire separates things and it cleanses things. The baptism by fire, and the cleansing that follows is seen as a process, not an event. The sanctification & transformation of the human soul are the product of a lifetime (and more--we believe in a pre-mortal existence and a post-mortal spirit world), not a single, powerful experience.
This cleansing and sanctification is designed to change our nature and to purify (see 1 John 3:2-3 & Moroni 7:48) such that on Judgement Day when we return to God's presence, we are prepared to stay in His presence.
Disclaimer: these thoughts are the product of my own study and do not constitute official statements by The Church of Jesus Christ of Latter-day Saints