I'd argue that obsessing about timeliness is of itself evidence of being high-strung or high-maintenance.
On-time is... sometime that day. Or perhaps tomorrow. Later than that is significantly late, but why worry about an hour or two? It really is a very, very particular western culture thing to be time-obsessed: it's not specifically something that all humans do intuitively.
That would be fine, and honestly I'd be much happier if my work was like that.
But my free time is limited and I have to be at work at 9am on Monday, no matter how long I wait for a selfish person to turn up. So I'm not going to sit there waiting for long...
A friend of mine moved from the East Coast to teach college in Idaho, and he made the comment that punctuality just wasn't a thing out that way. He's a historian, and he surmised that it was because Idaho never had its society fundamentally altered by industrialization and the time-keeping it required.
But yeah, I try not to get hung up on other folks' timeliness, unless there's going to be a knock-on effect. I know shit happens. I bring a book so I can put my time to good use if I have to wait. An obsession with punctuality in and of itself might be a sign that a person is optimizing what's measured rather than what's actually important.
On-time is... sometime that day. Or perhaps tomorrow. Later than that is significantly late, but why worry about an hour or two? It really is a very, very particular western culture thing to be time-obsessed: it's not specifically something that all humans do intuitively.