This is where the student, and many posters, are losing me. When I am on the phone with a doctor's office, they may say, "I have 8:30 or 12:00 available." I would then reply, "The 12:00 works best for me." Usually, the receptionist would reply, "OK, I have you down for 12:00." When OP said she had 8:30 or 9:00 available, why does that mean the appointment is made? The OP never said, "We will see you at 8:30 or 9:00", so why would anyone assume the appointment is set? I have never heard of someone offering two (or more) times meaning the appointment is made in any setting, whether it was a casual get together or a meeting with a teacher. Where from is this assumption coming? Genuine question because I truly don't understand why someone would make that assumption.
I can think of a couple different things. One is like wolfie said, that the rest of the message suggested there was no need to respond if one of those times worked for the student. I think if you're having to parse a message for one particular small word or phrase that changes the entire meaning, it was probably a little too ambiguous if you really needed a certain outcome.
Another is that the student did
call back and chose one of the offered times; the OP just hadn't gotten that message yet. In the student's response (from 9:30pm) she picked the 8:30am slot, so it would be rather silly of her to not
be there at 8:30am, even if she hadn't received a confirmation (IMO). There just wasn't time from when the student chose her slot, for her to wait for the OP to confirm it before showing up.
Also, IME some places don't really expect to give an explicit confirmation, especially if doing so necessitates sending yet another message (as opposed to a real-time conversation). "I have X or Y available," "I choose X," IME a lot of places wouldn't bother to respond but would expect you to show up at X, especially if there was a tight turnaround. It's kind of in the vein of whether you should send an email confirming you got the requested information and everything is okay, or if you should respond only if there's a problem.
Assuming attitudes were polite all around, I don't think anyone did anything rude. It's just that wires got crossed a bit. On the other hand, if the student had an entitled attitude--"Of course
I have an appointment at 8:30! What do you mean
she isn't here??" that's rude in and of itself. That would be rude even if the OP had confirmed her appointment, but Lisa was called away on an emergency and couldn't make it.