What are they doing for one hour that's so important that other people should wait on them - if I have a meeting at 10 and they aren't there, can I just leave to handle other work and show up at 12?
If they're late they probably have a reason for it (you might not think they have a good reason though, but who are you to judge that?) they're not doing it for the purpose of annoying you, even though you two might have a meeting you still aren't the centre of the world. So don't leave just to retaliate (because that is why you are talking about leaving and showing up even later) because that will accomplish nothing at all.
Once they arrive you can even tell them they're late, and they will apologize. Anger on the other hand will just leave you both miserable.
I don't know, it's slightly annoying when people are late, but I find even more annoying when people rant to no end about late people.
Also, one thing I'd observe is that there's a difference between social occasions and business. Even as an American, when my family says they're going to show up at 2pm, I don't sit there at 1:59 at the door tapping my foot. However, business is different; if you've mutually agreed on 2pm and you come in at 2:40 pm without having checked or communicated, there is a problem there in many cases. There may have been further scheduled events for 2:30 (odds are good you're meeting with someone who has lots of meetings, statistically), or whoknows what. The casual approach to time risks having three unrelated meetings that were scheduled to be separate trying to happen all at once.
Whether firm or casual time is abstractly "best" is a hard problem, but when it comes to business effectiveness I don't have a problem saying punctuality is a benefit to getting more stuff done. "What if I don't want more stuff done because even that is an American thing to say?" Well, surely we'd all rather then get our things done so we can hit the bar freely later? And it isn't an American truth that businesses really need to accomplish things to survive and thrive, it's the nature of the Universe we live in, where we must work for our sustenance, however distasteful you may find that.
On the face of it, I agree. However, I think this whole thread is neglecting an important dimension by only mentioning "the" time of the meeting. No one ever expects a person to arrive at the exact instant of the scheduled event. There's an interval implied by "the" time, and it's quite possible that the difference here is in cultural expectations regarding the size of the interval or its placement relative to the stated time (US Military culture, for example, seems to have an implied tolerance of -5 min / +0 min).
It may even be the case that your hypothetical second party thinks the first is a bit quirky for insistently describing the meeting as occurring at a particular time, when it is "obvious" to them that what is meant is "we'll meet this afternoon".
Edit: moving scare quotes ("the time" -> "the" time)
"Michigan Time — The tradition of starting every class, meeting or event 10 minutes late."
The way around this is for people to tell me to show up "around" 8pm or whenever.
I fully recognize that this is something peculiar to myself, and I would never begrudge anybody else for showing up at e.g. 2:04 for a 2pm meeting. But I would definitely apologize for being a few minutes late if I did the same.
It's neither an American truth nor the nature of the Universe, it's a story. Older than America (not the continent nor the people in it, but the idea), to be sure, but not that old. The necessity of work, indeed the celebration of it, is obviously interesting to discuss in the age of automation.
...in the West culture.
At what point does that become less reasonable than expecting other people to show up at the time they agreed to show up?