From someone who does janitorial work in a medical setting for a living...
The part about urine is accurate (clean). In that case, wipe down with toilet paper.
Feces or blood? You can wipe it down and probably be safe. If you want to be sure, then have some lysol/clorox wipes in your purse and whip out one of those to use on it. As long as you get the seat surface where you are touching clean, you should be good.
Or, just hit another stall. But yeah, the supervisor here typically has a batch of clorox wipes in her purse for when she has to travel outside the building, just in case (she IS a germaphobe).
As to HIV on any surface, if the body fluids have been exposed to air more than 10 seconds, it's gone, kaput, an ex-parr, er virus. It is not pining for the fjords.