Generally speaking you are correct but I would not say people left their own denominations to join the Pentecostal church, rather like all new denominations just some of the initial founders did, but as it grew it became its own proper denomination with its majority being its own converts.
It is interesting to look at the Holy Spirit under the heads that you have mentioned because if you put a wide-angel lens on the entire history, we could almost say the whole movement should have been expected as well as a kind of expected counter-movement.
Let me explain the high level history as I see it and this will give a good explanation of each of the term in context.
Starting with the Reformation (Luther, Calvin, etc.) it is true they taught the cessation of extra-ordinary gifts. The subject rarely comes up and when it does it is usually under the simple explanation as to why those extra miraculous gifts had ceased.
In the early Church the Holy Spirit was sent forth in visible form. He descended upon Christ in the form of a dove (Matt. 3:16), and in the likeness of fire upon the apostles and other believers. (Acts 2:3.) This visible outpouring of the Holy Spirit was necessary to the establishment of the early Church, as were also the miracles that accompanied the gift of the Holy Ghost. Paul explained the purpose of these miraculous gifts of the Spirit in I Corinthians 14:22, “Tongues are for a sign, not to them that believe, but to them that believe not.” Once the Church had been established and properly advertised by these miracles, the visible appearance of the Holy Ghost ceased. (Martin Luther, Commentary on Galatians 4:6)
Though Christ does not expressly state whether he intends this gift to be temporary, or to remain perpetually in his Church, yet it is more probable that miracles were promised only for a time, in order to give luster to the gospel, while it was new and in a state of obscurity. It is possible, no doubt, that the world may have been deprived of this honor through the guilt of its own ingratitude; but I think that the true design for which miracles were appointed was, that nothing which was necessary for proving the doctrine of the gospel should be wanting at its commencement. And certainly we see that the use of them ceased not long afterwards, or, at least, that instances of them were so rare as to entitle us to conclude that they would not be equally common in all ages. (John Calvin, Commentary on the Synoptic Gospels)
That they are not communicated unto any by a sudden afflatus or extraordinary infusion, as were the gifts of miracles and tongues, which
were bestowed on the apostles and many of the first converts That dispensation of the Spirit is long since ceased, and where it is now pretended unto by any, it may justly be suspected as an enthusiastic delusion; for as the end of those gifts, which in their own nature exceed the whole power of all our faculties, is ceased, so is their communication, and the manner of it also. Yet this I must say, that the infusion of spiritual light into the mind, which is the foundation of all gifts, as hath been proved, being wrought sometimes suddenly or in a short season, the concomitancy of gifts in some good measure is oftentimes sudden, with an appearance of something extraordinary, as might be manifested in instances of several sorts. (John Owens Works, Volume 4)
Yet this is not the whole story. Among reformers there is a wide range of emphasis of the spiritual life, awareness of the Spirit, necessity of the Spirit, etc. Among those who seemed always mindful of the Spirit in the following generations, among the Puritans in the tradition of reformed theology, were men like John Owen. Possibly the most extreme, among 'the reserved Puritans', was Jonathan Edwards, who claimed to experience a revival and outpouring of the Holy Spirit in his own church. After reading his accounts of that revival it seems very hard to deny it as nothing less than a genuine extraordinary work of God.
We now naturally find the stage for the Pentecostal movement in Los Angeles. The most insightful thing I have ever heard a preacher say about the Devil is this: First the Devil will do everything to keep you from fully committing your life to God, once you finally give everything to Him, the Devil changes his tactics and he tries to get you into extremes, to make you over-committed to one thing in order to neglect another. During the birth of the Pentecostal movement you had a few people zealously seeking God in prayer, in hopes that they could experience a revival, like that experienced by Edwards, and this was a good thing. However to add to this good zeal, a devilish doctrine was added which said no such revival is real unless accompanied by speaking in tongues. In other words, there was a doctrinal lie inserted into the movement at its root that basically did not recognize the biggest outpouring of the Holy Spirit ever since the gospel times, which was during the reformation!
The problem we face when trying to form an opinions about 'revivals' claimed by people like the Azusa Street Revival in Los Angeles, which gave birth to the Pentecostals, is that we feel they must be fully good or fully bad, but can't be both. We make this incorrect assumption because with Christ this was true. To deny the 'revival' in Christ and to accuse his works to be of the Devil, was blasphemy against the Holy Spirit. So it seems to follow that we must not question any 'great work of the Spirit' without putting ourselves under the same risk of danger. However, this is simply not true, even under the most amazing and genuine outpouring of the Holy Spirit, in the New Testament records, we find both the work of God and the Devil can run side by side. For example, the Corinthians seemed to have been experiencing the gifts more than any other church, yet the Devil was at work in them more than any other church also. (I am not assuming in the pentecostal revival that it was the work of the Holy Spirit as I was not there, but hypothetically assuming there were a mixture of sincere ad insincere believers, all intently calling upon God, some may have had genuine experiences with the Spirit while others with the Devil, or their own ecstatic psychological state, all at the same location.)
We now naturally expect a Charismatic movement. Imagine if you will, God meeting the answers of people prayers and truly manifesting his presence to sincere hearts going to Azuza Street. Now comes the cynic. Contrary to their expectation a minister goes to see what's going on, expecting to find a good reason to reject it, yet to their surprise 'boom' they fall under the presence of God and are amazed to find his love revealed to them and they renew their commitment to him. What is going on? how can this be? At the same time imagine the Devil pressing people to extremes, adding false teachings. Also imagine basic weirdos joining in, acting strange, bringing dishonor to God' part in it. Well what happens? What happens is some people want the Holy Spirit experience, but start to recognize the Devil, so there is an acceptance plus a backlash. This is the general charismatic movement.
After the charismatic movement calms down, some balance will be insisted like 'tongues is not the evidence'. Next messages like we must keep Christ the central message, not the Holy Spirit. Then God's sovereign will decides when there is a revival, you can't always expect it. Many have prayed for revival for years and it has not come, etc. All of this avoidance of extremes exists among many mainstream Charismatics. As the cycle completes, we find we are basically back at Jonathan Edwards as a state of total commitment and eager desire for the Spirit, yet without all the extremes that the Devil is so eager to push us into. However because the whole thing is a kind of theological whirlwind you will find good strong balanced believers scattered across all the denominations among many bad examples also crowding into each denomination. True some denominations may be far healthier than others, but each and all of us are in a deadly all encompassing war over our souls and faith transcends one's particular station in life and denominational biases.