A generator of a periodic pattern is a set of cells that is equal to the set of its parent cells one period ago.
An oscillator that has no other generator cells may be called stator-generated. If the bare stator is a precedessor of the oscillator, it may be called naturally stator-generated. A special case of this are oscillators where the rotor comprises only off cells in one of the phases (e.g. on-offs).
The glider's generator is the alternating L/Z tetromino; the same generator reappears in many c/4 diagonal spaceships, such as Orion or Swan (both times in some phases as a part of a larger polyomino).
A T-tetromino-based generator appears in many (most?) c/4 orthogonal spaceships.