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The term God or god refers to He who must be worshipped or that which is worshipped respectively.
He who must be worshipped is one in the sense that he is the only true God; any other god is not god but is being worshipped ignorantly.
Thus we have God the Father, the Son, and the Holy Spirit. Not God the Father, God the Son and God the Holy Spirit
This is probably just semantics as we're likely talking about the same thing but looking at it through a different lens.
I interpret God the Father, God the Son and God the Holy Spirit as three separate and distinct persons. ONE God but manifested in three different persons. But I do understand what you're trying to convey with "God the Father, the Son and the Holy Spirit". Thank you for your perspective.
Perhaps Proverbs 8:22-31 also provides insight into Jesus being present since the beginning of creation.