I think it is helpful to distinguish between linear **functions** (representing the relationship between independent and dependent variables) and linear **models** (representing the relationship between the model parameters and the outcome).

A linear model can be represented by a non-linear function. A linear regression model is any model that is represented by a linear function in the *parameters*. A polynomial can be represented as such. 

Likewise a nonlinear regression model is any model that is represented by a nonlinear combination of parameters. 

To quote from [Wikipedia](https://en.wikipedia.org/wiki/Polynomial_regression): 

> Although polynomial regression fits a nonlinear model to the data, as
> a statistical estimation problem it is linear, in the sense that the
> regression function E(y | x) is linear in the unknown parameters that
> are estimated from the data.