I don't know what they're smoking but I definitely want some.
1) "Space invaders" scenario is perfectly deterministic, they just have a problem with incorrect abstraction. In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears. Ditto for special relativity (it has its own problems with singularities, though).
2) Even if we allow the "space invaders" it still does NOT mean the system is not deterministic. If we know all the details about the system then we can predict its behavior with arbitrary precision. That knowledge, of course, must include all the infinitely fast objects. Laplace demon is still able to predict everything with arbitrary precision.