Not bad, though a bit verbose. I think the question may be a bit off for getting the best response. One of the more important properties, and what distinguishes them from the rationals the most, is the properties of them as a linear order (every bounded subset contains a least upper bound inside the real set itself). And I think something like containing a dense countable subset (the rationals) yields their unique/categorical property. This is mentioned in the book, as you know.

So you have these desirable properties your intuition wants them to have, and are able to prove that the description given uniquely captures it. So if two people were to discuss them, you can be sure that their conceptions are identical. There's a little nuance im hesitant to even mention, but I will, and that's that these canonical model proofs rely on second order logic. Which is fine, but when you get to completeness theorem it might confuse a bit since there you are restricting to first order logic.