Epistemically speaking, the existence of God is not axiomatic. Your second claims is more accurate, though not entirely. Knowledge of God's existence is derived from observed features of reality. However, these features are very general and not scientific per se; rather, they are presupposed by empirical science. Examples include the reality of change, causality (especially per se vs. what science is generally concerned with, per accidens), or the existence of things. The denial of these general features would undermine not just the possibility of science, but the very intelligibility of the world. You would hang yourself by your own skepticism.

These are also not axiomatically accepted features either (except perhaps in the sense that they are in relation to the empirical sciences, as science presupposes their existence).