Skew arithmetic circuits are usually defined a bit differently: in every multiplication gate, one of the inputs of the gate is a constant or an input of the circuit (we assume that multiplication gates are binary).

This is very similar to your definition, if *linear* is interpreted as *syntactically linear*. That is, one of the inputs of any multiplication gate is either a constant, an input of the circuit, or the output of an addition gate whose inputs are inputs of the circuit or constants. Using the distributive rule, we can take a circuit satisfying this definition and turn it into a circuit satisfying the usual definition.

Another interpretation would be that one of the inputs of any multiplication gate is a linear function of the inputs, that is, it is *semantically* linear. For example, $(a+b)^2-(a-b)^2$ is semantically linear but not syntactically linear. This is probably not the interpretation which is meant here.