There is an attempt to try to describe the theology of what happens when a person gets saved. And then there is the fact that on our part belief is how we connect to this salvation. You may say it differently but the overall truth is the same. Edwards talked about faith being imparted immediately by a divine and supernatural light. In another place he said that Christ has died and everything is waiting for you to come by faith. There are going to be places where our human minds can't do this at once so we have to say both things. And because we are saying two things, one will be said first and one second.
You don't want to minimize the fact that you have to believe. And you don't want to go with the idea that belief is totally an action of your own mind or something that you have conjured up on your own, to use at your convenience. Calvinists like Owen warned people not to make light of a calling or conviction because this can be sovereignly withdrawn and you will never be saved. Yet Owen said in other places something very similar to Boettner. So what do you make of those seeming inconsistencies? I don't think they can or really even should be resolved.
"Because" is a funny word. You definitely believe in order to be saved. That is your actual linking to or receiving the benefits of salvation. So it is legitimately a "condition" for salvation. But your belief is not the reason you were saved. That is due to the work of Christ so in that sense you are not saved "because" you believed.
I read a lot of Calvinists who were pastors and teachers as well as theologians and I don't find them to be as rigid in their theology when looked at practically as we see some people argue. I know that's not very satisfying in a theological debate setting but it is something to keep in mind, especially when debate heads toward calling each other false teachers or heretics. Although in fairness, the old guys didn't seem to have a problem calling each other names so I guess go ahead.