"Please discard paper towels into wastebasket" is direct and specifies exactly what kind of behavior is desired. Your example doesn't do that. It says there is a wastebasket, but the only request is not to throw paper on the floor. It could be satisfied by leaving the wet paper towels on the sink, leaving them draped on the door handle or slapping them on the wall. (How's that for passive-aggressive!)
The original message is both shorter and clearer. It's both pro-active and respectful. Getting angry at the notice and littering to show you won't tolerate "condescension" is just being a dick.