Would this be considered a responsive design?
Yes, you can and should create different designs for desktop and mobile. Otherwise, it would be just the same design scaled down (like with non-responsive websites). You can also hide advanced functionality or completely remove it.
From a technical standpoint you should also care about how these designs transition into each other (Breakpoints, layout grid). This will make it easier for the devs to develop something that scales from mobile to desktop and feels like one product.