Your reasoning about the electromagnetic field is right. I think the element of confusion here is the definition of nonlinear. Nonlinear in system theory means that the superposition principle is not valid.
From Wikipedia: "The superposition principle states that the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually".
The electromagnetic field in vacuum is linear, because it obeys Maxwell's equations and it is a solution of a set of linear equations. The output is always proportional to the input: you put a bigger charge and you get a bigger field. A linear combination of electromagnetic fields, with constant, real coefficients, is a new field which obeys Maxwell's equations, thus it is still linear and the superposition principle is still valid.