"I do not want you to die!"
Translation:Jag vill inte att du dör!
By main clause do you mean independent clause and by subclause do you mean subordinate or dependent clause? Because if that is the case, then this is what I think you are saying: If I translate the sentence as, "I do not want that you (should) die", then in English, "that" is a subordinating conjunction. This means that "Jag vill inte" is the independent clause (V2), which is exactly where "vill" is. So far so good. "Att" functions as a subordinating conjunction, which means that the V2 rule does not apply to "du dör". Therefore "dör" does NOT come directly after "att" and there is no inversion. Jag vill inte att du dör. Förstår jag?
Yes, totally. I think main clause/independent clause and dependent clause/subclause/subordinate clause are just different words for the same things.
Like you say, att is a subordinating conjunction. In a dependent clause, the V2 rule does not apply, but the subject must come before the verb.
If we compare sentences with subordinating/coordinating conjunctions, they can look like this:
COORDINATING Jag vet och du vet
SUBORDINATING Jag vet att du vet
Those two look like things are in the same order, so it may seem that it doesn't matter much which one is which. But then there's the BIFF rule – Bisats Inte Före Finit verb: In a dependent clause, inte must go before the finite verb. So if we add inte everywhere, we get this:
COORDINATING Jag vet inte och du vet inte heller
SUBORDINATING Jag vet inte att du inte vet
In the independent clauses, we still have V2 and inte gets to go after the verb. In the dependent clause, inte sneaks in before the finite verb.