many people are quite sure that (e.g.) a Goedel sentence for their favourite formalization of arithmetic is either true or false (and by the latter they mean not-true).

Those people seem a bit silly, then. If you say "The Godel sentence (G) is true of the smallest model (i.e. the standard model) of first-order Peano Arithmetic (PA)" then this truth follows from G being unprovable: if there were a proof of G in the smallest model, there would be a proof of G in all models, and if there were a proof of G in all models, then by Godel's completeness theorem G would be provable in PA. To insist that the Godel sentence is true in PA - that it is true wherever the axioms of PA are true - rather than being only "true in the smallest model of PA" - is just factually wrong, flat wrong as math.