Here is the question:
So, I am not asking for these definitions. What I am asking about any potential or actual definition of infinity is whether it is not a contradiction in terms and a self-defeating exercise to define what is indefinite?
The OP is not looking for a definition of infinity, but whether infinity can be defined. Also the OP is not interested in a mathematical definition or one which defines the infinite as the negation of finitude. Here is the OP's concern:
It may well be that the notions of God and infinity are very closely connected, and perhaps have a common history. This would then raise questions about the definition of God. Is it not self-defeating to attempt to define the transcendental, for the defined transcendental renders it immanent? In the same way, an instantiated infinity, an infinity one can refer to, is not infinite.
Dominic J. O'Meara provides a view from Plotinus of what is involved in returning to the One, Plotinus's notion of God. Our knowledge including our definitions are at best derivative. O'Meara cites the Enneads IV. 8. 1. 1-10: (page 104)
The arguments [that allowed us to return to our true self as soul] lead us even further. They not only show soul what it is; they also lead it to see that the knowledge it has is derivative, that it derives from a higher form of thinking, the divine intellect which, unlike it, does not need to work through long logical processes, but possesses knowledge in a different and superior way.
Whatever we have for definitions are derivative from this higher form of thinking, Furthermore these arguments must be left behind: (page 105)
It should be noted that if the transformation of our lives that comes with becoming intellect is facilitated by philosophical arguments, these arguments must also be left behind: they make use of logical processes and are not required in the perfect knowledge of intellect.
This suggests, as the OP suspects, that any definition of God (or infinity) may need to be left behind at least given the neo-Platonism presented by Plotinus.
O'Meara, D. J. (1995). Plotinus: an introduction to the Enneads. Oxford University Press.