In my field, political science, we always tend to just merge the GORs to create 4 English regions - North (NE, NW, YH), Midlands (EM, WM), South (EA, SE, SW) and London (LN). It's a clumsy definition, but it's one that nearly everybody can agree on. Most government work is done by the GORs and the 3 northern regions nearly match up with the 4 northern BBC regions, so it is a workable definition for the purposes of this thread.
But I imagine any definition of the North will face the same issues with it that we face with the GOR definition - there are parts outside the geographic north that feel more northern than some places in it - this YouGov survey
suggests that about 25% of people who live in the Midlands consider themselves part of the North. In the past YouGov have done by county surveys of what is the Westcountry or what are the Home Counties, but frustratingly haven't done one for the north. But it does show that you can't just draw a line on a map that dictates where the North starts - especially considering it is often in opposition to an even more vaguely defined South.
There is also the homogeneity issue - the North isn't in fact a homogeneous block. Politically, there are multiple 'norths' that behave differently, have different social attitudes and which sometimes aren't even geographically contiguous. The same is true culturally - with multiple northern cultures existing which are often quite different from the generic Northern culture. The North East and Merseyside, in particular, are noticably culturally distinct.
But this is the fundamental problem with regionalisation - without statutory definitions of what constitutes a region, you are never quite sure how small or large your regions should be. Choose large regions and somebody will suggest that you are ignoring the distinct subregions and say you are just recentralising rather than decentralising. Choose small regions and people will call you needlessly parochial and say you should be cost-cutting by merging together some of the regions. The debate over English regional government has been going on intermittently since the 1960s and still nobody has agreed a solution. I doubt this BBC shake-up will be the last.