Is this definition of a "complete graph" correct?
I'm not sure what "official definition" you have in mind but your definition of a complete graph is correct: it implies that every pair of distinct vertices are connected by an edge.

At least, it does assuming that by "connected", you mean "has an edge to". It's best to avoid using the word in that way, since "connected" in graph theory refers to the existence of paths that might be longer than a single edge.