[soVeryTired]

In my experience those debates are usually between experts who deeply understand the difference between ABC and XYZ widgets (the example I'm thinking of in my head is whether manifolds should be paracompact). The decision between the two is usually an aesthetic one. For example, certain theorems might be streamlined if you use the ABC definition instead of the XYZ one, at the cost of generality.

But the key is that proponents of both definitions can convert freely between the two in their understandings.