I don't feel there's much difference between e.g. the Coq way where types are used to bake specifications + proofs into the program and the Isabelle way where you write a program with less complex types and prove properties about the program after the fact.

General mathematical proofs? If you mean paper proofs they wouldn't be considered formal as they don't go down to the axiom level.

I guess that makes sense to count axioms as strongly typed.