Yeah, it's tricky. I try not to get into nuances that the reader cannot be reasonably expected to understand based on what came before. I always say the posts are an exercise in carefully designed ambiguity rather than careful precision.
However, I do not feel it would be productive to replace "continuous" with "complete" in this instance. One of my biggest annoyances with mathematics is its insistence on redefining ordinary words with ordinary meanings to something very specific and counter-intuitive. Case in point, I was taught that left and right are the same "direction" mathematically. This is only going to make people look at mathematicians as if they're crazy. It also ruins simple definitions like "two vectors are equal if they have the same length and point in the same direction", which is perfectly clear using the ordinary meaning.
If I correctly understand the concept of a continuous function defined on Q, it is in no way intuitively 'continuous'. Rather, it satisfies the epsilon-delta concept of getting arbitrarily close, which is necessarily done in discrete steps. If you then go on to define functions that are continuous only in a finite set of points, you end up with what I can only call "troll-math". It's fun to think about, but it has no place in an introductory text. Even Wikipedia starts out by saying "the function is continuous if, roughly speaking, the graph is a single unbroken curve with no 'holes' or 'jumps'." A function defined on Q should not be called a curve, in my opinion.