Is there any formal axiomatized definition of Determinism written in First Order Logic (FOL), with a semantic interpretation maybe rooted in Set Theory, or based on any other logical system and semantic interpretation, for that matter?
One form of determinism would be a Turing machine.
Turing machines have a formal definition.
Turing machines cannot decide every language. For example the Halting problem is undecided. However, if you generalize the notion of a Turing machine to have an infinite number of states, you can get around this and recognize any language: Consider a deterministic countably infinite "automata": with one state for each possible input string (it would look like an infinite binary tree). You can configure the accept/reject states however you want to recognize any language you want. This doesn't have a corresponding Turing machine because it has an infinite number of states.
If you generalize the notion of a "language" to be any set with higher cardinality, you could "construct" (basically "hard code") a corresponding "automata" that "decides" elements of that set. You would have a start state; each element of the set would transition to an accept state; other elements not in the set would transition to a reject state. This basically says determinism captures the notion of a boolean function.
Therefore anything you can define logically has a corresponding deterministic "automata" that decides it.
Therefore determinism is equivalent to logic.
The question you are asking, is determinism not a metaphysical belief system?
The answer is it is a metaphysical belief. The definition of determinism in summary says,
Determinism is the philosophical theory that all events, including moral choices, are completely determined by previously existing causes. (Wikipedia)
As soon as you use the term "all events" if there is just one event that is not deterministic, the model fails. And this would require infinite knowledge and the ability to see everything in the past and the future. Such beliefs are outside the scope of science and are therefore metaphysical.
Ofcourse believers would love an absolute proof of their belief system, to conquer the non-believers, but that is the nature of faith.