I'm given a model of the form $E(\ln Y|X)=X\beta$.
Should I call it a linear model or loglinear model?
I'm assuming $Y$ is log-normally distributed.
$ E(\ln Y|X)=X\beta $ is a linear model of transformed variable $\ln Y$, more precisely, it is linear in parameters. As I noted in the comment, you can introduce additional variable $Z = \ln Y$ and then $ E(Z|X)=X\beta $ is just an ordinary regression model. You can make all kinds of transformations to your variables, but when talking about linearity of the model, we have in mind the relationship between variables.