In my grad stat classes, "Linear Regression" usually includes things like my example above - i.e. linear functions of the (potentially transformed) explanatory variables including higher order terms. Is this different from the how the term is widely used?

If you say W = XxX, then make a model that's linear in W, it's a linear model. If you use both X and XxX, I don't think there was a definitive answer... until Wikipedia, of course. Which says no.