There are several things that the Bible can mean by the phrase “Holy Spirit”, but none of them suggest that it’s God Himself being referred to...First, contrary to the allegations Christians made that we Muslims are being silly when we say that in the Koran the Holy Spirit is Gabriel, the Bible itself sometimes refers to angels as “spirits of God”:
And between the throne and the four living creatures and among the elders, I saw a Lamb standing, as though it had been slain, with seven horns and with seven eyes, which are the seven spirits of God sent out into all the earth. (Revelation 5:6)
Read through the rest of the relevant chapters of Revelation and it will be clear that these seven spirits are the seven angels in 8:2 and so on. So the term “spirit of God”, which Christians consider synonymous with the term “Holy Spirit”, can mean “angel”. The word “spirit” can also mean “prophet”, as we can tell from 1 John:
Beloved, do not believe every spirit, but test the spirits to see whether they are of God; for many false prophets have gone out into the world. (1 John 4:1)
Hence the Islamic belief that the Holy Spirit prophesied in John 14-17 is Muhammad (peace be on him). Then, of course, there is the literal meaning of the phrase...“breath”:
And when he had said this, he breathed on them, and said to them, "Receive the Holy Spirit." (John 20:22)
Then there is the more nebulous meaning of the term, in which the spirit is simply the spirit of inspiration, in the ordinary English sense of the word. Christians lump up all of these different meanings, all of which are both obvious and obviously different, under the same definition, and that is precisely why the Holy Spirit is the hardest part of the Trinity to define, the most meaningless of the three phrases. There is no single meaning of the phrase in actuality, but only a long string of different non-Trinitarian meanings, but Christians consider them all to be the same, and so they end up with a term that cannot be clearly defined, or can be only when everyone has a different definition to give. It’s jargon, in other words, nonsense talk. That’s what you get when you oversimplify a complicated series of different meanings of a term--you inevitably end up with a difficult, nebulous, hard to fix, subjectively interpreted meaning on your hands.