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.
Your example of a tabular layout is a perfect illustration of the need to have separate layouts for mobile and web devices.
Tables invariably impose horizontal + vertical scrolls on mobile devices — which is usually challenging to use. One a mobile device, a list view is far better representation of tabular data.
Contemporary programming environments enable quick configuration. For instance, on Angular, the
BreakpointObserver service  allows developers to deliver screen-size-based breakpoints, show & hide elements, and toggle between different layouts.