Oops. So many definitions intertwined to create an interpretation consistent with our understanding, culture and belief, that anything is possible. For example, your interpretation of a wave does not coincide at all with mine. A wave is a disturbance that transmits energy, and is not something that "waves" around. The passage of that wave or disturbance is what generates the oscillations that most tend to interpret as a wave. Thus, every time a wave passes, the electric or magnetic field, or any entity that interacts with that wave, oscillates. Since it is not possible to identify exactly what the wave is, we limit ourselves to mathematically explaining the oscillations that are generated when that entity interacts with a wave. For this reason, we tend to confuse oscillations with a wave, when in reality they are nothing more than the consequences of the interaction with a wave. We have no idea of the very nature of true waves...
Yes indeed, defining a wave is really tricky.
Although, anyone can go down to the beach and recognize waves.
I am sure you will agree that defining light as "a wave" is, at minimum, serious shorthand, and more likely a completely fallacious statement.
It will really help when we start working with the aether again. Then, we can at least define in what media the wave is waving in.