Is this definition of a "complete graph" correct? Is it correct to say that:

"A complete graph is a graph in which each vertex is connected to all other vertices in the graph"

That's how I always thought about it, the official definition is different (I know that).
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.