What is the precise definition of a nonlinear signal?
In my opinion, the term nonlinear signal doesn't make much sense. If you refer to nonlinear signal processing then this is not about the processing of nonlinear signals, but about the nonlinear processing of signals, i.e., about the use of nonlinear systems.

A system $\mathcal{T}$ is nonlinear if it doesn't obey the superposition principle. I.e., if $y_1(t)=\mathcal{T}\{x_1(t)\}$ is the response to input $x_1(t)$, and $y_2(t)=\mathcal{T}\{x_2(t)\}$ is the response to input $x_2(t)$ then

$$\mathcal{T}\big\{\alpha x_1(t)+\beta x_2(t)\big\}=\alpha\mathcal{T}\big\{x_1(t)\big\}+\beta\mathcal{T}\big\{x_2(t)\big\}$$

is generally not satisfied.

Well-known examples of such nonlinear systems are median filters and Volterra filters, but there are many trivial operations, such as taking the magnitude of a signal, which must be classified as nonlinear processing.