I think the only reason the Borg were part of the First contacted as Seven has described it is because of what they first did when they got there. They shelled the crap out of the settlement, and had it not been for the Enterprise destroying the Sphere, there would have been no launch anyway.
Geordi and Riker were the two sitting inside the Phoenix with Cochrane during the flight. We know from Lily pestering Zefram right before the initial attack that she wasn't going to go up in that thing with a drunken pilot, that had it not been for what happened, she would have been there with him. But what about the third seat? Although it is never said, its most likely the third person was killed or maimed in some way during the attack, but since the Enterprise crew needed to repair whatever damage had already been done by the attack, they ensured the flight went accordingly.
My theory, as there is no way to prove or disprove this whatsoever (unless speaking with the writer), is that the Prime Universe originally went off without a hitch, and the flight went how it was supposed to with all three original crew of the Phoenix. When the Borg went back in time to assimilate Earth, the Enterprise had to fix what was broken. Chances are some differences did occur, although marginal compared to what their present time was before the events of the movie.
As far as the Mirror Universe is concerned, at some point in Earth's distant past, a choice was made that drastically changed something and turned it into an alternate universe. According to Dr Michio Kaku's books: Parallel Worlds and Physics of the Impossible, it is theorized that this would be part alternate universe and part multiverse theory. The fact that there are a million different choices that can be made in any instance is irrelevant to us, as we have already made the choices that brought us to here, but had any one of us done something different, it is plausible to assume that this conversation may not have ever occurred in the first place. From whatever point it may be, it's theorized that those different choices have already happened simultaneous to what we in this universe did, and because all the choices were made, the alternate universe already exists.
The only time, according to Kaku, something becomes an alternate timeline, would be when past events in our universe are changed. If someone goes back into the past and say, kills Hitler before he took power, and tries to come back to his present, he would create an alternate timeline from his own and would not be able to return to his timeline, as he has changed history. The fact that the Borg did what they did in ST:FC changed the timeline, and the Enterprise tried to fix it. Whether it worked completely speculative at best, since it's impossible to know everything that could have changed from their original timeline, but technically, even the Borg's and Enterprise's presence in the past could have changed the timeline without them having done anything. So for Seven to say the Borg played a large role in first contact is true from her perspective, as she would be a product of the "altered" Prime Universe, but if it had never happened, who is to say exactly?