I don't see how a hard fork is somehow not Bitcoin. It's simply an upgrade. A bigger upgrade, but an upgrade nonetheless. If it is done in the face of serious controversy and there is a real split, then there is a naming problem. Otherwise it's just as much Bitcoin, just a new version.

It could be Bitcoin version 2.0 for example. If there is a permanent split, then the other network would be called something else.

I'm going to side with the version where the block size limit is raised, that is for sure. If it is done in a smart way though, it's a big change and needs to be thought of carefully. I don't advocate "just raising it", I advocate planning the change well and thinking about the long term future as well.