The term Lax pair is fairly often used in a broader sense as a pair of linear equations (not necessarily of the form you mention, i.e., $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$) whose compatibility condition is the system under study. In fact, this notion is occasionally stretched even further, for example, in the theory of integrable dispersionless systems one encounters nonlinear Lax pairs, see e.g. here and references therein.

The term Lax pair is fairly often used in a broader sense as a pair of linear equations (not necessarily of the form you mention, i.e., $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$) whose compatibility condition is the system under study. In fact, this notion is occasionally stretched even further, for example, in the theory of integrable dispersionless systems one encounters nonlinear Lax pairs, see e.g. this paper and references therein.