As you said, if the father vetos it, it is not a marriage.
So what is this "marriage" that the father might veto? That is the definition I'm looking for. You have provided conditions under which a marriage must be formed, or may be denied. But that does not define the word "marriage".
Looking at it another way, imagine yourself as a missionary walking into a village in a non-Christian culture, getting to know the people around you, and working out who is married to whom. What criteria would you use? You'd really do that quite instinctually, without going into the detail of who slept with whom back when they were virgins. I'm just trying to verbalise that.