Your own doctor knows the answer to this - different practices could have different definitions - and should have asked you questions that tested their own criteria. These might be things like "are you bleeding?" and "did you lose consciousness?" along with "how bad is the pain?" and "how long have you felt this way?" After these questions they might say "call an Ambulance", "go to A&E", "we can see you at 2:30" or "we can see you 6 days from now at 2:30."
Asking you "is it an emergency?" is a foolish question when you don't know their criteria. [People ask foolish questions all the time, and forget that not everyone knows what they know.] So answer by providing the information you would use to make the decision yourself: "I think so, it has come on very suddenly and the pain is worse than I've ever experienced" or "not enough to call for an ambulance but I can't stop it bleeding and I really would like it looked at today" or "I suppose not, I just want to be sure it's not the start of something serious". Just saying "yes" doesn't give them any reason to agree with you and schedule something sooner or send you to more urgent care.