The Holy Spirit in relation to Father...

Another Trinitarian question which I have been searching for an answer for years now.

2 Cor 3 talks about the Holy Spirit.

v3 says: "You show that you are a letter from Christ, delivered by us, written not with ink but with the Spirit of the living God, not on tablets of stone but on tablets of human hearts."

v8: "how will the ministry of the Spirit fail to be even more glorious and splendid?"

Now in 2 Cor 3:17 it actually says that the Spirit is Father and not that Father merely sends the Spirit: "Now the Lord is the Spirit, and where the Spirit of the Lord is, there is liberty"

For me v17 indicates that the Holy Spirit isn't a separate entity, but Father's own spirit which He sends out. This might be difficult for us to understand as our spirit is bound to our physical bodies till death. But is this true for a spirit being?

I was taught that the Holy Spirit is a separate entity as it is seen in the form of a dove coming down on Yeshua, but that scene description might just be a rudimentary description by the writers to describe a wave of light/spirit moving over/onto Yeshua at baptism.

It might still have been Father Himself being seen in His spiritual form (as if He has any other form...). Hence why Father's face is always "hidden" within a light. That "light" is Father's visible form.

What are your thought on this?