A purely logical TM would be understood to enumerate different proofs depending on the model of the axioms you used to specify it. This is how there can be one model of PA where such a TM halts, and another model of PA where such a TM doesn’t halt. So your plan doesn’t do the work you seem to think it does.
Don’t think of this as “there is one actual thing, but it mysteriously has multiple behaviors.” Even though it’s really convenient to talk that way (I did it above just now), maybe try to think of it like when you pick some axioms, they don’t actually pick out a single thing (if they’re complicated enough), instead they’re like a name shared by multiple “different things” (models), which can behave differently.
A purely logical TM would be understood to enumerate different proofs depending on the model of the axioms you used to specify it. This is how there can be one model of PA where such a TM halts, and another model of PA where such a TM doesn’t halt. So your plan doesn’t do the work you seem to think it does.
Don’t think of this as “there is one actual thing, but it mysteriously has multiple behaviors.” Even though it’s really convenient to talk that way (I did it above just now), maybe try to think of it like when you pick some axioms, they don’t actually pick out a single thing (if they’re complicated enough), instead they’re like a name shared by multiple “different things” (models), which can behave differently.