I think you're inconsistent as fuck, and it's causing your kids, and other adults in your life, problems.
You say you told your child to sit down several times. Then when you realised he was interacting with another child, ignored the behaviour and let him do it. To your little one , (and your OH) - it's very much 'if I do it enough times, mummy will give up and I can do it anyway'.
Your OH saw you give up/change your mind on teaching your son to sit in his chair, likely assumed you were giving in/fed up, and intervened because your son wasn't behaving appropriately, nor listening, and to everyone else at that point, you gave up parenting for a moment.
Boundaries and rules should be consistent so that the child understands them - otherwise the child learns to try the behaviour because they haven't learned from you whether it's actually ok or not. They don't have the cognitive ability yet to suss out its ok 'because it's a booth chair and he knows the other child'. That isn't really fair to the child to be honest, it's confusing! Nor will your partner know where the boundary lies, and that's because of you and your inconsistency in parenting. I also think it's a daft reason to make an exception to the rule that seems to make sense mainly to you, that it was a booth chair and the people in the next seating had a child, that may or may not know yours.
It's also really rude to let children stand/climb on furniture. Child friendly or not, kids shouldn't be climbing up on chairs to be bothering the people in the next booth. You likely don't care, but it does make you look rude, and your OH was reacting to this as much as he was to your child. I wouldn't want a child in the next booth standing up to interact with us as we're trying to eat a meal. Hello's and interactions are fine, in an appropriate manner - ie. on the ground, when you're not in the middle of eating in a public restaurant.
If you're with a man, who has met and is going for meals out with your kids, I assume it's because you are serious about him, in which case, one day you will be close enough that he can be involved in parenting. It's fine that today wasn't that day.
That being said, if I was him I'd run a mile. People who let their kids put their feet/stand all over seating where people sit to eat, and bother people in booths nearby, appear utterly embarrassing and disrespectful IMO - I'd be put off by this kind of thing, as much as you've been put off him. I couldn't be dealing with the inconsistency either. I'd be off.
It's really hard for people to date single parents (and vice versa). They're your kids, the other person is 'taking them on' in a sense, at some point down the line, but not allowed to have a proper say. Often an impossible scenario.
I'm going YABU, because I think you brought this situation on and it could have been avoided if you parented appropriately. He went too far and should have asked you to seat the child properly, and if you wouldn't, he should have just left. That's what I would have done.