The language of interaction is based largely in quantum field theory. In that context, the quadratic part of the action produces the propagator of the field and any cubic or higher parts left over are lumped together and called the interactions. This is because these extra terms form the interaction vertices in Feynman diagrams.
From this point of view, the connection to linearity of the equations of motion is simply the observation that upon varying the action to produce the equations of motion, the quadratic terms will produce contributions to the equations of motion. On the other hand, cubic and higher terms in the action result in quadratic and higher contributions to the equations of motion...non-linearity.
From a classical point of view things are less clear because you would first need to define what you mean by "interact." One thing we can do however, is take the non-linear terms in the equations of motion and group them together so the resulting equations of motion look schematically like (linear)=(non-linear). If we want to lean on "intuition" from, say, Maxwell's equations, we can think about the right hand side as being like a source current (like charge distributions and currents in Maxwell's equations). It just happens to be one that these "sources" depend upon the fields themselves. If you like, you can then view this as the statement that the fields in a non-linear theory act as sources for themselves, and hence they will react to their own presence.
That's a bit of a stretch, but as I said, it really depends upon how you would like to define "interaction" in a purely classical context and how hard you're willing to lean on the ideas that come from quantum field theory.
I should note that the reason quantum field theory has light by light scattering, and this is a purely quantum effect, can be understood in several ways. If you like looking at diagrams, it should be a simple exercise to convince yourself that there are no tree-level diagrams with only external photon legs. To connect two external photon legs, the lowest loop way to do this would be to have an electron loop in the middle. Loop diagrams do not contribute to classical scattering processes, hence light by light scattering in QED is a genuine quantum effect.
Another way to see the same thing is to note that the quantum effective action for any theory is, at tree level, just the classical action. For QED, this means that higher order (cubic and higher) interaction terms start at the 1-loop calculation of the action. So if you were to try and write down the equations of motion for the vacuum expectation of the fields in QED, you would need to consider quantum corrections to the action before encountering a non-linear term in the equations of motion.