It's an interesting question that I imagine might have a different answer now than back then. Having read countless accounts of immense dust clouds and dust six inches deep on the march, I wouldn't think a soldier would keep his cup outside or uncovered. I've done it before on just a couple mile march and wouldn't do it again.
Then again, these guys were getting their water from a stream or pond, not a water buffalo. They could easily dunk it and rinse it off. While reenacting I'm not of a mind to waste water in my canteen or from the spigot to rinse off a cup to fill it, so I generally store it in my haversack. Not having one has never come up, as no group I have ever been with will allow someone on the field without a canteen.