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I wonder why this distinction is not introduced in elementary mathematics when students are taught about angles. Perhaps it wouldn't be simple!

Teaching students that numbers like 0 degrees and 360 degrees are coordinates of angles instead of being angles would relieve the teacher of having to double-talk about them being "different, but really the same" angle. Yet there are situations when zero degrees and 360 degrees denote different things. For example, a moving object making a "turn of 360 degrees" is different than its making a "turn of zero degrees".

Perhaps teaching angles in a way that made sense would involve teaching both equivalance relations and coordinate systems.