What are gravitational waves in general, not just in the weak field/linearized theory?
Have a look here
I'm not sure that the issue is really the weak field approximation. Isn't the real issue something that occurs in other contexts, e.g., plain old E&M on a flat background? Say I have a space that's empty except for a point charge. In the rest frame of the charge, this is a static field. In a frame where the charge is moving, an observer at a particular point will see a time variation in the field, and will probably interpret this as a wave disturbance passing by. Indeed, according to this observer, energy is being transported by the field. However, there is no energy being transported to infinity.
For gravity, in spacetimes with special symmetries, I guess one way to go is to use the Petrov classification. This is analogous to defining (pure) EM waves as null fields (although there are exceptions).
If the spacetime is asymptotically flat, I believe you can compute the Bondi news function to decide whether there is radiation.
I doubt that there is a definition that works in general. To call something a gravitational wave, you want it to be something that's transmitting energy. But in GR we don't have a definition of the energy of the gravitational field, except in special cases such as asymptotically flat spacetimes.
Separate names with a comma.