I've never heard the word 'lacto-ovo vegetarian', but that doesn't mean it doesn't exist (it does now, anyway!) - I would understand it (though I wouldn't be sure whether the lacto-ovo referred to eating/not eating milk and eggs).
A vegetarian eats nothing but vegetables; some people don't eat meat but do eat fish - but a vegetarian I once knew didn't bother with qualifying the use of vegetrian by using a prefix; he just called them 'not proper vegetarians'. (I even know someone who sometimes eats fish, depending on the brain size - she's a marine biologist, so she knows things like that. There isn't a word for that sort of vegetarian.)
A vegan goes several steps further than a vegetarian. A vegan doesn't use anything to with the killing of animals. They don't use leather shoes or woollen clothes. They don't drink cow's milk (because although the cow isn't killed - at least, not immediately - milk can't be supplied without a farming system that involves killing calves). They don't eat some sweet foods that are processed with animal products.
But I'm not aware of any specialist vocabulary for this particular area.