Newton Game Dynamics :
Our engine implements a deterministic solver, which is not based on traditional LCP or iterative methods, but possesses the stability and speed of both respectively. This feature makes our product a tool not only for games, but also for any real-time physics simulation.
as for PhysX I don't know, but most of the popular physics engines use iterative solvers (bullet, havok) your 'one stage' definition matches an iterative solver.

As for the 'two stage' defintion then you're looking at, as mentioned above, LCP based solvers (Linear complementary problem). LCP is a broad term for certain classes of co-dependent mathematical problems (think quadratic equations) such as for instance, physics solvers.

So an iterative solver achieves a higher degree of accuracy through progressive refinement (think tight collisions between 3 or more objects) where are a non iterative solver (not that I know any specifics) uses various mathmatical tools developed for thermodynamic problems to, I assume, produce close approximates as an alternative.

In the words of Newton (game dynamics, not the man) I should imagine that the best physics engines use a wide combination of cross discipline techniques.