I just assumed a first offense because the police gave you a summons rather than arrested you.
In any case, there would be no extra cause for concern with a case like this. Actually there are two public urination statutes here in NYC. One is in the administrative code -- that's the one with just civil penalties. The other is a violation of the Health Code. Both are treated more or less like misdemeanors.
With either one you're likely to wind up with a small fine $25-$50 and either the ACD or, if you can't get that, a Disorderly Conduct. A Dis Con in NYS is not a crime. It's a violation, the criminal court equivalent of a traffic ticket. It looks better than "public urination" on a record if you can't get the ACD.
Are you aware, incidentally, that California will make you register as a sex offender
for public urination if you're convicted of it over there? Something you might want to keep in mind when you get back home.
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