> It seems "constant-time" isn't used to mean "the time taken is O(1) regardless of the size of the input n", but rather, "for a given input size, the algorithm is carefully written to do the same amount of work no matter what the specific bits of the input are, to defeat timing attacks".

This seems very obvious to me. "Constant" independent of the input size seems literally impossible for any nontrivial algorithm. At the very least you need to somehow read in all the input, and that's already input-size dependent.