Apparently public urination can overlap with indecent exposure, depending on the situation, and the latter can result in a sex offender registry.
You hear the concept a lot, because it's a cover - "I'm on the sex offenders list, but I was just peeing. Didn't realize it was a playground, whoops!" is different from the true "I raped a girl, and despite the incredibly high burden of proof in that charge I was convicted, and now I have to tell you about it because I moved next door."
Look on the registry yourself - you won't see anyone on there for peeing in public: https://www.nsopw.gov/
What burden of proof?
This is a crime where you can be accused of having committed it several years in the past, with no supporting evidence of any kind, and convicted for no other reason than that you give someone a "rapist" vibe.
https://www.law.umich.edu/special/exoneration/Pages/casedeta...