I'm aware that the TSA does not decide this, and that's what I tried to explain to her; that every airline can decide on their own.

I still feel like I have no idea what the right answer is. For example, Delta says:

SO WHAT EXACTLY IS ALLOWED?
Approved Personal Items
1 purse, briefcase, camera bag or diaper bag
or 1 laptop computer (computers cannot be checked)
or 1 item of a similar or smaller size to those listed above

"Similar size" No actual measurements listed.