If I am correct, the linear regression model is said to be a model where the response is linear in the parameters. That allows the response to be nonlinear in the predictor variables.
I was wondering if there is a name for the model where the response is linear in both the parameters and in the predictor variables, so that people will easily understand its difference from the linear regression model?
The expression $X\beta$ is linear in both $\beta$ and $X$.
As a result, linear regression is already linear in both parameters and the predictors that are in the $X$-matrix.
What they may not be is linear in is the original predictor variables, if the $x$'s can be transformed.
For example, consider the linear regression in this question, where the $x$-variable is the time of year ("ToY", as a number scaled to be between 0 and 1). The regression isn't linear in ToY, but it is linear in the predictors that were actually entered into the regression, the various $\sin$ and $\cos$ terms.
In short, there's really no distinction; linear regression is linear in both $X$ and $\beta$. If you want to differentiate between transformed and untransformed predictors, it would be by talking about them in those terms (as being untransformed from the original variables).
