To be precise, a model is said to be a linear regression model when it is linear in its parameters. For extension, a model is said to be non-linear when it is non-linear in its parameter (Wooldridge, 2010, p.397).
As such, a linear model can have non-linear variables. A standard example is the Mincer equation, where wage is a linear function of education, experience and experience squared.
In your example, as your intuition tells you, parameters $\rho$ and $\beta$ are multiplied themselves, making it a non-linear model. However, your model is a rather special kind of non-linear model because its parameters can still be recovered using linear methods. In fact, you do not need non-linear methods at all. Your model is over-identified as you can test whether the estimated combined term is consistent with the individual estimation of each parameter or not.