I used the term ‘truth’ as used in mathematical logic. In a given model a statement can be true or false. Under a given axiomatic system a statement is either provable or not. We don’t use the word “true” when dealing with statement under an axiomatic system. We do use the word when dealing with a statement in a given model.

Your third paragraph doens’t make sense. In first order logic a theorem is a statement that is true in all models and is one that is provable. This is a result of the Completeness Theorem.