I think the issue here is not that this couple has a different parenting style but that the other mom thinks she can parent your children for you - pressuring your vegetarian kid to eat chicken? scolding them for eating a plum when it's not the designated "snack time" for her kids?

A different parenting approach could have involved her telling her kid she can't join in on the plum with your kid because it's not her snack time. Trying to stop your kid from eating her snack is really overstepping.

Also, you can't be that controlling when you accept free babysitting - that's essentially what they did by sleeping in late. Sure, there are things I've prefer my kid not have, like juice, but I can't flip out over it when a neighbor is doing me a favor by watching my kid for free while I step outside for a few minutes. Your "friends" don't seem to get that.

Maybe they usually have their kids sleep in later and found it annoying that you rounded them up early.

My mom has the rule that the kids are not allowed out of their rooms before nine and if awake cannot make any noise in their room. She also makes all the grandkids have a quiet hour in their room after lunch.

What happened is that you vacationed with people you're not compatible with, with whom you never established ground rules about who would pay for what and schedule stuff.

Things that could have gone differently:

1. Clear house rules - if the house rules are that everyone has to leave the kitchen clean after using it, or that everyone has to make their bed, or whatever, that should be spelled out before going to the beach.
2. Flexibility about food - what to feed them for breakfast, what is available for snacks
2a. Menu planning for the week - everyone has input, everyone shares in the cooking/clean up duty, everyone's food issues (your DD's vegetarianism) are taken into account.
3. Clear communication about discipline - if you're sharing childcare responsibility, I think it's reasonable that whatever adult is "in charge" is the one who gets to make discipline calls, within whatever standards are agreed by the parents. I wouldn't be in any way okay with the other parent spanking my child for an infraction, since we do not do that in our family. I would also be unwilling to spank, even if they do it in their family, for the same reason.
4. Clearer schedules - did you know that your hosts would be sleeping in for 3 hours every day?

Ultimately, what I decided was that I would not go on vacation with any family unless I knew them really, really well and could be reasonably sure that none of these things would be an issue.