Yes it does. Puppy killers like to claim that it doesn't. /s
Here are a few examples with names: https://www.menshealth.com/trending-news/a19541024/you-might...
There is a ton of posts from lawyers confirming the fact that in many states you can get convicted for public urination and put on the sex offender list. For example, here is a post from a Texas-based lawyer: https://www.craiggreeninglaw.com/can-you-really-become-a-sex...
The reason it's so hard to estimate the real number of these cases is that the crime is listed as indecent exposure or lewdness.