Are there absolutely no examples of cases where mathematicians disagreed about some theorem about some not-yet-axiomatized subject, and then it turns out the disagreement was because they were actually talking about different things?

There is such an example -- rather more complicated than you're describing, but the same sort of thing: Euler's theorem about polyhedra, before geometry was formalised. This is the theorem that F-E+V = 2, where F, E, and V are the numbers of faces, edges, and vertices of a polyhedron. What is a polyhedron?

Lakatos's book "Proofs and Refutations" consists of a history of this problem in which various "odd" polyhedra were invented and the definition of a polyhedron correspondingly refined, until reaching the present understanding of the theorem.