There is such a distinction to be made.

However I personally found it very helpful in real analysis to be able to take any question about the reals back to how it relates to this construction. This greatly helped my intuition.

IMHO, though, "an equivalence class of Cauchy sequences" is not the most intuitive representation of a real number. (A point on a line is; even Dedekind's cuts well-described in older editions of Baby Rudin are more intuitive.)

My perspective differs.

Dedekind cuts have a special case at all of the rationals. Which is weird. And the whole construct the completion using equivalence classes of sequences construction is one you'll encounter a bunch of times in topology. So it is worth learning it properly.

Plus if you're into that kind of thing, you can also construct the p-adics this way. :-)

Exactly. Moreover, a construction can be seen merely as a proof of the consistency of the set of axioms (i.e. it is the axioms rather than any particular construction that define what the reals are).