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The problem is that they have a number in mind, but refuse to state that number.

My parents regularly get into arguments about how many of an item were purchased as my mother will never use a number. It is always some descriptor word like "few", or "several" or "plenty."

Only after things are purchased will it be revealed that plenty means 16.

This begs the question because for me "a couple of X" is equivalent to "two of X". It's not an imprecise term for me. (The one exception that comes to mind is "a couple of things" which I understand to be just loosely defined all around.)

But would you say a bicycle has a couple of wheels?

Does yours have more than a couple of wheels?

Yes, I would.

This is my thought exactly. If I meant only ever exactly 2.00000000000 I'd say "two". If I'm saying "some small value of approximately two" I'll say "a couple".