Which is correct terminology: "A Cartesian plane" or "The Cartesian plane"? (As in the directions for a section of homework being, "Plot a point on ______ Cartesian plane." In that context, I feel that one of the two should be used consistently, but find there's little agreement on Google for which is appropriate.
I think from a philosophical point of view, both terms may be used. From an educational point of view, I would generally use the term "the" Cartesian plane in school context as every two Cartesian planes are isomorphic and should most of the time be considered the same object. This may reduce cognitive complexity. Only in cases where you have several distinct objects that might be called Cartesian plane, I would change the terminology towards "a" Cartesian plane. This could happen in a university course on geometry, where you may cut out Cartesian planes from projective spaces, for example.
Why do you think they should be used consistently? I think there are situations appropriate for each of these choices.