As you can see, different people have different opinions about what a "Diophantine equation" is. E.g., for a long time, some people meant rational solutions to equations, and others meant integer. These are significantly different, since, unsurprisingly, classification of integer solutions is much more complicated than classification of rational solutions.
And, indeed, whether asking for integer solutions or rational, the variously more-or-less restrictive types have different properties, as we know by this year. For example, I think all of the modern-times results that use (modern) algebraic geometry can only bear on polynomial equations, for obvious reasons, whether addressing rational or integer solutions. (Mordell-Weil, Faltings, et al).
But in a broad sense, especially if asking for integer solutions, a broad sense of "Diophantine" can certainly allow "reasonable" operations on integers. The question of whether there is any systematic machinery that might conceivably bear on such questions depends enormously on what, if any, larger context such a question could be imbedded.