MM — you’re smart. The possessive implies that the Holy Spirit is subordinate to God.
It’s more about specifying which Holy Spirit one is talking about.
Lots of people claim that what they are doing is from the Holy Spirit, but while they can claim it all they want, it doesn’t mean it’s the Holy Spirit as revealed in Scripture.
Same as for those who claim to follow Jesus, like the Mormon’s who teach another Jesus than the one revealed in Scripture.
Besides, subordinate does not necessarily mean unequal in essence. Even Jesus was subordinate to the Father’s will. He talked about it Himself. He Himself said that He came to do the Father’s will. That is being in subordination to the authority of the Father.
Does no such thing.
Is not God one but three persons? Thus couldn’t one say “God’s Father”, God’s Son, and God’s Holy Spirit? Or must one say God the Father, God the Son, and God the Holy Spirit?