Actually, the terminology and ontology are astonishingly simple. Electromagnetic waves are ripples in the electromagnetic field, which is a collection of 4 real numbers attached to every point in space and time (the four components of the "4-vector potential"). Gravity waves are ripples in the metric tensor field, which is a collection of 10 real numbers attached to every point in space and time (the independent components of a symmetric 4x4 matrix). The laws of gravity and electromagnetism are a differential equation satisfied by these components governing how they fluctuate from place to place and time to time.

Photons are merely quantized ripples in the electromagnetic field (according to quantum mechanics, the momentum and energy in these ripples come in discrete chunks).

It's instructive to wonder what could possibly be a simpler ontology. Perhaps one imagines a "medium" like that of sound, consisting of little atoms of some sort spread throughout spacetime. But then immediately this would beg way more questions: what are the atoms themselves made out of? When neighboring "atoms" are displaced why and how do they affect a given "atom" in order to propagate disturbances? What other properties might these atoms have and what would such a word as "properties" even mean since they are supposed to be the elemental constituents of light/gravity itself?

According to modern physics, none of this: just 4 + 10 = 14 numbers, no substructure whatsoever.

(Of course, there are other fields too, like matter fields and gluon fields and one a necessary part of the whole story is a declaration of what these numbers look like according to different observers, but these features are just elaborations on the basic picture of a collection of numbers attached to events.)