The red herring principle[1] is unfortunately popular enough in mathematical terminology to have a name and a page about it.
Roughly, a fooish bar will frequently be something like a bar except fooish, so not actually a bar. (Algebraic integer, multivalued function, manifold with boundary, etc.) On the other hand, a nonfooish baz when baz is normally fooish often means a not necessarily fooish baz, so a particular one might be fooish but we can’t assume that. (Noncommutative ring, nonassociative algebra, the very field of noncommutative geometry, etc.)
Roughly, a fooish bar will frequently be something like a bar except fooish, so not actually a bar. (Algebraic integer, multivalued function, manifold with boundary, etc.) On the other hand, a nonfooish baz when baz is normally fooish often means a not necessarily fooish baz, so a particular one might be fooish but we can’t assume that. (Noncommutative ring, nonassociative algebra, the very field of noncommutative geometry, etc.)
[1] https://ncatlab.org/nlab/show/red+herring+principle