Linear Model for Linear Regression
Theoretically, you do indeed have a linear model, yes.

You have a linear relationship between the dependent variable and your parameters.
You would still be going through the process of performing linear regression, fitting a line through points via a linear combination of some regressors. However you have a non-linear equation due to the higher-order regressors you have manually inserted (x1x2 and x2^3).

This is enough to still call it a linear model generally speaking - see some useful answers in this thread.

In general I wouldn't say you have a linear model in the strictest sense at the end, as you are modelling linear relationship between your dependent variable y and non-linear combinations of your regressors: x, x2, and x3, but perhaps there could be an underlying feature to be observed which is exactly equal to x1x2, call it x4, and then you would have removed one of the non-linear covariates. Tyhe same goes for the final non-linear term.