Yes it is confusing! They seem to be making it up on the hoof, as the research gets done, and politicians prevaricate and then panic.
The Pfizer 3rd dose and the booster are the same strength I think (Moderna booster is half dose). So I guess you’ve probably had to wait longer for your ‘3rd’ dose as it was on the ‘booster’ timescale rather than the 1-2 month timescale for the third dose? I think the important thing is that you have had it, and just recently too. Fingers crossed you are one of those people who do get a reasonable immune response, as seems this can be variable. Immunity typically starts to build up from 2 weeks, so at least the timing is good for this new surge. But do take care. Assume you have a weakened response and get others around you to take the level of precautions that you want to feel safe.
I hope you manage to get the NHS to sort out your status and they now get you marked down correctly for an additional booster.