Q&A: How do I react/respond to God, Jesus and the Holy Spirit differently?
The functions of the members of the Godhead are specific, and yet there is NOTHING that one member of the Trinity does that the other two are not also involved with. That includes the sacrifice of Jesus on the cross. Much of what actually happens to us personally (physically and spiritually) is through the agency of the Holy Spirit. He is the One who indwells us, witnesses to us, empowers us, convicts us, seals us, gives us spiritual gifts, and brings forth fruit in our lives. And yet, the Holy Spirit is called "The Spirit of God" and "the Spirit of Jesus." They are One.
As far as prayer goes, I think you need to be careful not to overthink this thing. The distinctions within the Trinity are highlighted for the purpose of our study and understanding, but we must be careful not to make too much of them. We worship and serve ONE GOD and addressing Him ought to be consistent with that reality.
You are correct in saying that in the Word we are told to pray to the Father, in Jesus' Name and in the power and leading of the Holy Spirit. However, we are not forbidden in the Word from addressing any Person of the Trinity. Once in awhile, I will pray on a Sunday morning asking the Holy Spirit to open our eyes to the Word, but it would be just as appropriate to ask God to send His Spirit to do that work among us. On other occasions I have prayed, "Jesus, open us to Your Word so that we can see what You want us to see." Again, the WORK of opening our understanding is ALWAYS through the agency of the Holy Spirit, but WHO we address specifically isn't really important. Because God is One, not just in purpose but also in essence, we are addressing ALL THREE at any given time.
I hope that helps. God bless!!