The way this was presented to me was that to some employers, being laid off for any reason was considered a bit of a stigma, so by resigning, I could truthfully state (and the company would confirm) that I had left of my own accord.
Not 100% sure what consequence this would have had on potential unemployment insurance claims, because I was about to emigrate anyway and not interested in looking for other work. So I chose to resign.
While I am not sure, I think legally here is a very grey area as employment information is protected, but I'm not 100% sure.
On the other hand, many jurisdictions have laws which make it pretty difficult to fire people for cause without clear evidence of misconduct or performance improvement plans and probationary periods for underperformance, so there's a good chance the termination threat is just a bluff to see who will resign...
I don't think anyone has ever contacted my previous employers, despite me having supplied their contact details as a reference numerous times over the years.
Companies are discouraged from saying anything more than the facts of employment ("We can confirm that Bob was employed as a software developer between these dates...") these days, so there's not much value in the new employer bothering as far as I understand.
Laws are getting tighter too. It’s now illegal in many parts of the US to ask you how much you made in your previous job when that used to be a fairly common question.
Sadly, many companies report your salary to the credit reporting agencies (or a tangential agency).
Not sure anyone with a brain actually cares about it, but they do ask about it if you apply for a job.