I don't think anyone is making mathematical errors in the actual model, we are just using different words which makes it impossible to communicate. If you dereference my words in your model, you will see errors, and likewise the other way.

Yep.

Is there a resource where I could learn the correct terminology?

My conventions for describing mathematical objects comes from a somewhat broad range of experiences and I'm not sure I could recommend a specific resource that would duplicate the effect of all of those experiences. Recommending a range of resources would entail learning much more than just a few conventions for describing mathematical objects, and you may not feel that this is a good use of your time, and I might agree. I can at least broadly indicate that some useful mathematical subjects to read up on might be real analysis and topology, although most of the content of these subjects is not directly relevant; what's relevant is the conventions you'll pick up for describing mathematical objects.

Sometime soon I might write a Discussion post about mathematics for rationalists which will hopefully address these and other concerns.