Lol, are you under the impression that escorting and providing sex for her "time" is legal? Sorry to disappoint you buddy (and maybe you better be more careful, eh?) If an escort actually only provided her time and conversation, then that would be and is a legal service. There are actually escort services which do only provide company for events, dates, etc. However, if you arrange a "date" with an escort and if you make no explicit statements or agreements about sex, and you go to a hotel room or something, and you bring a wad of cash, and then you engage in any number of sexual activities, and then you leave, she can be arrested for prostitution, and you for soliciting. Try telling the cops and judge you were only paying for her time. But chances are you will not get arrested, but not because what you were doing was legal, but because it's not cost-effective (or really possible) to police the entire internets.
The very simple, and really, only, reason that escorts get busted less frequently than street hookers has nothing to do with the legality of you paying for "time" vs paying for "sex" in your communications; it's merely that it's so much easier and less time consuming to prove the case when you have the explicit agreement as opposed to implicit. Why bother going to all that trouble to prove an escort was a prostitute when you can more easily and cost effectivly prove that a prostitute is a prostitute? I mean, this is vice, they're not dealing with murderers, and any fines levied aren't going to bring in that much money anyway.
So back to the original point, the definition is completely irrelevent since the menu and signage clearly stated prior to their ordering that the gratuity would be added for a party of their size, thus, it was clearly conveyed that the addition was not optional. This was not vague or misleading or prone to misinterpretation, nor did they claim afterwards that they didn't know the charge was mandatory, they simply refused to pay it. They ordered, and then they ate, which acts as "acceptance" of the offer at the price given.