What “God” means to me could be entirely different to what it means to you. For some people, God is a word that describes an older bearded white man. For others, God is considered more like a higher power or universal life force. It doesn’t really matter what you believe, because prayer is about invoking or summoning your vision to become manifest in the world. Whatever belief works for you, works.
As an intercessory missionary, I can’t say I agree with your definition of prayer. Prayer is personally grounded, an appeal to Someone. If you don’t have that, I think you’re not talking about prayer.
But once you do assent to that, then the question becomes “which Someone?” To whom shall I pray? And even in Christ’s example– especially in Christ’s example– we see not a manifestation of self, but an appeal to the Father for a manifestation of His will.
Christ taught us what prayer is, why we should do it, and how we should do it.