[It's not a different/separate/distinct person from his Father.....] - No, the Spirit of God has subsistence & is NOT the Father. 

1 Cor 2:11 ... Even so no one #knows the things of God except the Spirit of God. 

Jn 15:26 But when the Helper comes, whom I shall send to you from the Father, the Spirit of truth who #proceeds from the Father...

Jn 16:12-13 I still have many things to say to you, but you cannot bear them now. However, when He, the Spirit of truth, has come, He will #guide you into all truth; for He will not #speak on His own authority, but whatever He #hears He will speak; and He will tell you things to come.

Mat 28:19  in the name of the Father and of the Son and of the Holy Spirit

2 Cor 13:13 The grace of the Lord Jesus Christ, and the love of God, and the fellowship of the Holy Spirit, be with you all

Eph 2:18: For through Him we both have access by one Spirit to the Father.