The 'bay' is the rectangular bit and nothing else.
Unless there is established case law on this point (in which case: as you were), I fear quite a few county court judges would take quite a lot of convincing that a hatched area immediately around a disabled bay, which exists for the purpose of facilitating disabled access, does not constitute part of the bay.
I'm not saying I disagree with the argument, just that I think quite a lot of judges might.
Although if the only images the parking company has are those already provided, they might not be able to demonstrate much.
The points around the signage are very valid. The convoluted terms notwithstanding, there's that much crammed onto the signage that the key terms are very much not prominent!