A textbook in tensor calculus written by a one of my professors is full of statements that seem to skip important details, so I'm usually skeptical about certain claims that I find in his book but not in other books on differential geometry (the author is a physicist, not a mathematician).
One statement in particular is a real head-scratcher for me. Following the example very similar to this one: Manifold with different differential structure but diffeomorphic, the textbook states:
Even though the two different atlases are incompatible they produce compatible differential structures. That means that incompatible atlases can produce equivalent differential structures.
From the answers to the linked question I know now that these two differential structures are indeed equivalent in the sense that there is a $\mathcal{C}^\infty$ diffeomorphism between them, but I still belive those differential structures are in fact incompatible, because by taking a union of the two maximal atlases we still have two charts from the provided example that have a non-differentiable change of coordinates map.
Am I right in finding writing in this textbook really sloppy or is it that I just still don't understand something about compatibility of atlases?