For Szabo, I'm inclined to say 1968 makes most sense. That Romania added a year at some point in the 70s to make her of age for 81 worlds, before knowing about the rule change.
The main evidence for 1969 is Bela. He's a known unreliable narrator and has an incentive to say it didn't happen on his watch. The documentary could've been an error, or accurate repetition of a lie they were told for whatever reason, to show off or whatever.
Balanced against that, Szabo herself has always said 1968, for the whole of the time she could speak freely afaik. It seems too much pointlessness for her to admit the fake DOB but keep lying about what it was, alongside Romania changing it for no good reason since even a 1969 DOB would've made her LA eligible. There was no reason for them to change it as late as 1981.
I know quite often false DOBs don't make much sense. But this is multiple odd things for no reason, or Bela being full of shit. Ockham's Razor would suggest Option B.