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A bit of both, really—the term 'working class' is used by many of the commentariat to describe a particular type of person; those people may well identify as working class, but for purposes of political analysis they might more reasonably be placed in a different category dependent on their material circumstances. These material circumstances might more effectively explain these people's voting behaviours than a limited and reductive definition of 'working class'.
Again, I don't want to deny people's identities as such—there are obviously more dimensions to class than just the economic.
On the second point, I guess there has been a shift of sorts. Older generations of working class people who grew up in and around the labour movement will have different conceptions of class consciousness than younger people who are subject to similar, albeit different, labour market pressures in the present day. This (mostly) younger generation of workers who largely aren't homeowners in some cases might not consider themselves working class, in spite of the fact that materially speaking this is the role they inhabit. Of course, this is complicated—if you're from a middle class background you don't simply 'become' working class by virtue of your position in the labour market at any given moment, but still I would argue that these material circumstances are very important to consider, and are often effaced by the narrow definition of 'working class' that is so prevalent.
Sorry for the long and probably inconclusive post! I also don't think a relatively simple material analysis such as this can explain on its own why the result in Hartlepool was so bad.
Just to be clear, are you saying that the contortion you refer to is an inaccurate one perpetuated by more progressive types (i.e., your average metropolitan liberal elite) or an actual shift in the demographic of those who self-identify as working class?