I think it's a state of mind rather than a set of rules.A question I've thought about at times. My earliest 'Birmingham' ancestors were from Harborne and Northfield pre 1840s but I doubt they considered themselves Brummies.
As we get towards the late 19th century there are relatives born Aston, Balsall Heath and a few other places whose birthplace is listed as Bham (and some not) but how much that was due to them or the recorder I don't know.
I wonder if people whose families came from other parts of the country (and beyond) were more likely to consider themselves a Brummie when strictly speaking they were born in Aston etc. As opposed to people whose families had lived in that area for generations.
Judging it strictly on birthplace then I wouldn't be, as I was born in Marston Green Hospital.
And do people in Sutton Coldfield consider themselves part of Bham? I think some/most would like to think they weren't.