I did a little more digging (emphasis on little), and I found only one real life example of someone having to register as a sex offender after peeing outside, but it requires a massive amount of unproven belief of the guy in question.
According to Juan Matamoros, he was required to register as a sex offender (and eventually convicted of not doing so when he moved to Florida) for just peeing in public.
However, to believe that, you have to take him at his word and not look to deeply at the actual case. In reality, he wasn’t just convicted of public urination, he was actually convicted of “open and gross lewdness”. The newspaper (and I) were unable to obtain the court/police records, but it does appear from a docket report that Juan spun his “just peeing” story at trial to the judge, but was convicted of the worse offense despite his story. In addition, he was sentenced to 2 years in prison for it, which would be shockingly severe for just urinating outside.
So, if you believe the word of the defendant who was convicted of a sexual offense, sent to jail, and registered as a sexual offender that he was just peeing outside, you have one example. However, I don’t believe him, and I’ve been unable to find any other real life examples.