"Points on the line" is fine for the first, second, ..., tenth cut at a definition. Sure, completeness is the biggie for the reals compared with the rationals, algebraics, etc.

Still, as in the OP, mentioning Dedekind cuts is okay since it is one way to establish completeness, but there is much more, e.g., as in

John C. Oxtoby, Measure and Category.

and even that doesn't fathom all that is special about the reals. E.g., for just a little more, there is the continuum hypothesis, that little thing!

The OP wants to say that by mentioning Dedekind and completeness he is getting at what the reals really are; no, instead he is just cutting one layer deeper of something that has likely some infinitely many layers available.

Yes, yes, yes, I know; I know; the reals are the only complete, Archimedean ordered field, okay, after we have defined completeness, Archimedean ordered, and field and explained why these are important.

So, back to "points on the line" -- it's actually pretty good for a first cut.