By using the term "Logos" the author of John's Gospel intends to explain the Lord Jesus Christ in terms of Greek philosophical ideas current at the time.
He may have done this because, in the Greek culture for hundreds of years beforehand, introducing a new god was heresy, it was strictly frowned upon: In Athens for some time it was a capital offence. Thus, famously, Socrates in 399 BC was sentenced to death:
the accusers cited two impious acts by Socrates: "failing to acknowledge the gods that the city acknowledges" and "introducing new deities". (https://en.wikipedia.org/wiki/Trial_of_Socrates)
Whenever some new eastern religion came along the Athenians would see if the new god was actually one of their accepted gods but just with a different name: Such was OK. But to introduce an entirely new god was entirely unacceptable.
The Apostle Paul, in opening his message to the Athenians, on Mars Hill, masterfully got around this problem by saying he was just going to tell them how to better to worship one of their own gods... "the Unknown God" (Acts 17:23). John gets around it by saying Jesus Christ is their "Logos". If that is the reason, then it was another masterstroke to open the Greek-influenced mind to the reception of the Gospel: it would also explain why his gospel starts the way it does. "Be receptive to what I am about to tell you in the coming manuscript because I am not introducing any new god for I know that would be unacceptable.
Or he did this, not because Greek philosophy influenced his beliefs, but rather to make the Gospel easier to understand for those who knew the Greek philosophy; he used Greek philosophical teaching about the Logos to act as a bridge to faith in Christ as the Son of God for some of his readers.
To know the background to those ideas might add to our appreciation of John's Gospel.
This is a quote from a book by C.H. Dodd:-
Augustine, in a well-known passage of the Confessions (vii. 9), writes:
Thou didst procure for me through a certain person... some books of the Platonists translated from Greek into Latin.
There [i.e. in those Platonist books] I read - not in so many words, but in substance, supported by many arguments of various kinds — that
in the beginning was the Word, and the Word was with God, and the Word was God.
The same was in the beginning with God.
By him were all things made, and without him was not anything made.
That which was made in him was life, and the life was the light of men.
And the light shines in darkness and the darkness comprehended it not.
And that the soul of man, though it bear witness of the light, is not itself the light; but the Word of God, being God, is the true light that lighteth every man that cometh into the world.
And that he was in the world and the world was made by him, and the world knew him not.
But that he came unto his own and his own received him not, but as many as received him to them gave he power to become sons of God, even to them that believe on his name, I did not read there.
Again I read there that God the Word was born not of the flesh nor of blood, nor of the will of man, nor of the will of the flesh, but of God.
But that the Word was made flesh and dwelt among us I did not read there.
Quoted from "The Interpretation of the Fourth Gospel" by C.H. Dodd (page 10 of book, which begins chapter 2 of Part I, "The Background"):-
Reading on from that page 10 is a long section too long to give here.
Or you can look at Augustine's Confessions book 7 chapter 9, page 170, online here:-
For those who would like to read paragraphs 13 and 14, chapter 9, book 7, of Augustine's Confessions more exactly:-
"And Thou [Lord], willing first to show me how Thou resist the proud, but give grace unto the humble, and by how great an act of Thy mercy Thou had traced out to men the way of humility, in that Thy Word was made flesh, and dwelt among men:— Thou procured for me, by means of one puffed up with most unnatural pride, certain books of the Platonists, translated from Greek into Latin. And therein I read, not indeed in the very words, but to the very same purpose, enforced by many and divers reasons, that In the beginning was the Word, and the Word was with God, and the Word was God: the Same was in the beginning with God: all things were made by Him, and without Him was nothing made: that which was made by Him is life, and the life was the light of men, and the light shines in the darkness, and the darkness comprehended it not. And that the soul of man, though it bears witness to the light, yet itself is not that light; but the Word of God, being God, is that true light that lights every man that cometh into the world. And that He was in the world, and the world was made by Him, and the world knew Him not. But, that He came unto His own, and His own received Him not; but as many as received Him, to them gave He power to become the sons of God, as many as believed in His name; this I read not there.
Again I read there, that God the Word was born not of flesh nor of blood, nor of the will of man, nor of the will of the flesh, but of God. But that the Word was made flesh, and dwelt among us, I read not there.
For I traced in those books that it was many and divers ways said, that the Son was in the form of the Father, and thought it not robbery to be equal with God, for that naturally He was the Same Substance. But that He emptied Himself, taking the form of a servant, being made in the likeness of men, and found in fashion as a man, humbled Himself, and became obedient unto death, and that the death of the cross: wherefore God exalted Him from the dead, and gave Him a name above every name, that at the name of Jesus every knee should how, of things in heaven, and things in earth, and things under the earth; and that every tongue should confess that the Lord Jesus Christ is in the glory of God the Father; those books have not.
For that before all times and above all times Thy Only-Begotten Son remains unchangeable, co-eternal with Thee, and that of His fulness souls receive, that they may be blessed; and that by participation of wisdom abiding in them, they are renewed, so as to be wise, is there. But that in due time He died for the ungodly; and that Thou spared not Thine Only Son, but delivered Him for us all, is not there.
For Thou have hidden these things from the wise, and revealed them to babes; that they that labour and are heavy laden might come unto Him, and He refresh them, because He is meek and lowly in heart; and the meek He directs in judgment, and the gentle He teaches His ways, beholding our lowliness and trouble, and forgiving all our sins. But such as are lifted up in the lofty walk of some would-be sublimer learning, hear not Him, saying, Learn of Me, for I am meek and lowly in heart, and ye shall find rest to your souls. Although they knew God, yet they glorify Him not as God, nor are thankful, but wax vain in their thoughts; and their foolish heart is darkened; professing that they were wise, they became fools."
Augustine is at pains to show that, though the Greek philosophers speak of Logos and even of the Son of God they never speak of this Son becoming man and dying for sinners. That is unique to the Christian Gospel.
C.H. Dodd's work is referenced here:-
Philo of Alexandria (20 BC to 40 AD) was the (Jewish) philosopher who sought to construct philosophical bridges between Hellenistic philosophy and Jewish religious thought. In Chapter 11 ("The Doctrine of the Logos in Philo's writings" by Professor C. H. Dodd) we read:
The pivotal and most developed doctrine in Philo's writings on which hinges his entire philosophical system, is his doctrine of the Logos.
What advantage did John see in using the term Logos? If Augustine is right (see his quote relating to John 1:1 above) then by the time of the writing of John's Gospel a developing theme in one/some of the schools of Greek philosophy included a progress from a unitarian understanding of God to a Two-Persons-One-God, a "Bi-une", understanding. In this school of Greek philosophy the Logos is not just "God's word", but rather one of the Persons of the Godhead. So John used the term to help his readers who knew Greek philosophy: it is a much smaller step of the understanding to go from a Biune God to a Triune God than from a Unitarian God to a Triune God.
The OP writes:
Some Christians I've spoken to reckon that all that Greek philosophy is irrelevant and that we should only focus on the "Christian" meaning of the word, which is simply "Word", nothing more or less than that.
Whatever the reason(s) John started his gospel with fourteen verses associating our Lord Jesus with the Logos of Greek philosophy, it does not undermine the ability of those without a Greek philosophical background to understand his message. And so, those Christians are more right than wrong: what matters is that unlike the Greek "Logos", the Word of God became flesh, died for our sins, and rose again the third day.
And it is those differences between our glorious Word of God, the Son of God, and the Saviour of sinners, and the Greek Logos which really matter.
We don't need to delve overmuch into why "Logos" was used by digging down and down into Greek philosophy. Its only the Bible that really matters.