From the Missouri sex offender registry website...
Offenses Requiring Registration (589.400)
Any felony offenses in Chapter 566, RSMo, Including attempts and conspiracies.
Any offense in Chapter 566, RSMo when the victim is a minor.
Offenses not under Chapter 566, RSMo that require registration are:
Kidnapping when the victim was a child, pursuant to Section 565.110, RSMo - excluding kidnapping by parent/guardian. (589.400.6)
Felonious restraint when the victim was a child- when not the parent or guardian
Promoting prostitution in the first, second, and third degree
Sexual exploitation of a minor
Promoting child pornography in the first and second degree
Possession of child pornography, furnishing pornographic material to minors
Public display of explicit sexual material
Coercing acceptance of obscene material
Promoting obscenity in the first degree
Promoting pornography for minors or obscenity in the second degree
Incest
Use of a child in a sexual performance
Promoting sexual performance by a child
Endangering the welfare of a child, pursuant to 568.045, when endangerment is sexual in nature
Sexual contact of sexual intercourse with a resident of a nursing home
Genital mutilation of a female child
Furnishing pornographic material to minors
Abuse of a child when such abuse is sexual in nature
Juvenile offenders, certified as adults, who have committed an offense comparable to or more severe than aggravated sexual abuse
Juvenile offenders fourteen years of age of older who are adjudicated of an offense that is comparable to or more severe than aggravated sexual abuse
I don't see anything indicating that urinating in public would constitute an offense that would require registration in Missouri. I'm no attorney, but I think it would be difficult for an prosecutor to convince a judge that anything about urinating was sexual in nature. I'll research your laws more later, have to get back to work.
WM