talking about someone who was ill, I'd say that they had to go to hospital

talking to one of my medic friends about what time their shift started or asking what time someone's appointment was, I'd ask them what time they had to be at *the* hospital.

This all makes sense. Another USn way of saying, "I was in hospital", (and the one saying that knows which hospital he was in) would be "I was hospitalized". Probably I'd advise someone attempting to learn the vagaries of English to follow the Commonwealth usage. But how far do you carry it? The summer I worked at Arlington Park Race Track I would speak of "going to the track every day" but this is the same idiom a racing fan would have used. Of course, in both cases, a particular track was meant, so maybe it isn't that far from the Common(wealth) usage.