Is an engine surge classed as an engine fire?
If the fire is momentarily and stops by itself: No, this is not an engine fire. Flames can come out of the engine at both ends in a compressor stall (also called surge - watch the movie which is linked in the question!), but that is no engine fire. See this question for another example where spectacular flames develop, yet the engine is not on fire.

If the fire can only be stopped by shutting off the fuel supply or the engine itself: Yes, that is an engine fire. I guess you refer in your question to this bulletin on PW4000s, where the No. 4 bearing overheated due to insufficient lubrication. In that case the oil itself overheated and burned, but this still makes it an engine fire.