You are right, and that would be the most natural way of defining H to be a subgroup of G. But requiring for all that is just an equivalent, and apparently some think more useful way of defining the same concept (more useful in the sense that proving a subset H of G to be a subgroup requires proving only that single statement).

Assuming that funny definition of being a subgroup, if you set you get that ; thus H contains the neutral element e. And once you have shown that you can set and get that H also contains the inverse element of every one of its elements: .