The first thing I learned as a new DM was that it doesn't really matter what the rules actually are, as long as everyone's on the same page. I heavily homebrew everything about our 5e games to the point that a lot of it is unrecognizable in the rulebook.
My players know this; we've had several discussions about how we're not married to the rules, and we're more interested in having fun with the game than following the book to a T. We've thrown a lot of rules out the window and I've made on-the-fly calls for situations where I didn't like what the book was offering me. And it works for us!
Part of the key here is communication; they know that I'm not necessarily following the rules they know from the book. However (and this is the other key part) they trust me. We've established that I have the ability to make up rules as I see fit. They also know that a) those rules apply to both player characters, monsters and NPCs alike, and b) if they genuinely have a problem with a ruling I'm open to discussion.
Essentially, the important part is to establish consistant rules, stick to them, and leave yourself open to criticism/change if the changes to the rules aren't working for you or your players. Also be willing to admit if you make a mistake! Often players appreciate you being up front and will be okay with a small bit of retcon to fix the problem.
It's the consistancy that makes this system work; if this NPC is marked dead but is making saving throws, every NPC should be making saving throws unless there's a compelling reason otherwise. In your case I might have made an effort to make it more obvious that the NPC was attempting to stablize (potentially difficult to show without metagaming, but possible) so that my players had a chance to realize that this was something that's possible in my world.
Your players see only what you show them, and it's easy for them to feel cheated if you change the rules behind the scenes and then surprise them with it. I don't believe there's anything wrong with fudging the rules to bring the NPC back to life. The issue is that you blindsided your players with the change. Admit you made a mistake pronouncing them dead, apologize (briefly) and establish that going forward either a) this was a one-time thing as that NPC shouldn't have died, or b) NPCs will always have the ability to recover so they know to look out for it.
TL;DR: You made a small consistancy error and your player felt blindsided and probably lashed out because of that. It's not a huge deal in the grand scheme of things; apologize, establish what your reality is now, and make your best effort to telegraph appropriately going forward. There's no harm in fudging the rules as long as everyone's aware that you can/will fudge when it's in the players'/story's/characters' best interests.