It's not a union of those classes, it's a different class, and as you say, it's more powerful, because it can be projected (my reduction adds additional variables) into 3-SAT and SAT instance.
Yes, 2XSAT is the name I gave it, and I couldn't find it anywhere. The reduction is surprisingly simple, yet nobody mentions it. That's why I am warning people here - just based on this alone, I 80% believe that P=NP with a practical algorithm (which either way involves solving linear equations). And I wouldn't be surprised somebody coming up with the algorithm.
The reason why I say it's an intersection is because that's how the set of solutions of an instance looks like. That's what we need to figure out - how to characterize the sets of solutions described by SAT instance (i.e. sets of assignments to boolean variables that satisfy the instance).
However, it's not that easy, even if you characterize them as interesections of 2-SAT and XORSAT instances, set of solutions to 2-SAT is notoriously hard to characterize too, for example, #2SAT is not known. And polynomial algorithms for 2-SAT and XORSAT are doing very different things, and it's not at all obvious how to generalize them into a common algorithm that can do both.
