Drill
A “drill” is defined as a spinning device that has an adjustable chuck able to clamp down onto a twist bit of just about any size. Yes they still make them – usually called “drill/drivers”, because you can also put screw bits into them. The simplest ones available are usually variable speed (drive screws carefully) and reversable (back out a bit, or unscrew a fastener), the better ones have both drill and screwing mode with a numbered release chuck to stop you from driving fasteners too deeply. You can get cheap ones and expensive ones; you can even still easily find ones with a cord rather than with batteries. Most chucks today do not have “keys” to tighten, but are called keyless chucks. Just search for “drill/driver” or “corded drills” if you don’t want batteries. In a quick search I see corded drills from Black&Decker, DeWalt, Makita, Genesis, Bosch at all kinds of prices.
Impact Driver
An impact driver is a special category. It “usually” does not have an adjustable chuck (which can slip easily with the impact action), but has a hex socket, most commonly ¼ inch hex – which fits almost all fastener heads. There are even twist drills with a ¼” hex base to fit into these machines. They will often have a choice of actions – simply spinning (drill) or hammering in the direction of the twist (hammering the screw action).
Hammer Drill
Then there are “hammer drills” that hammer forward to force bits into masonry. Many are multi purpose – simple screw drive with one or two speeds and a release chuck, impact drive for getting larger screws or lag screws secured and hammer drive for masonary. Some can stay stationary, or turn, for the hammer action allowing use with masonry or even wood chisels. These are usually larger tools for heavier work.
Combo tools
My old Makita multi-drill has all of these with the standard symbols – from the right to the left: Drill, Impact driver (hammer hitting sidewise), Hammer driver (hammer hitting forward) and screw at 2 speeds – both screw speeds responding to the numbered release clutch.
So in answer to your question -- all the old drill functions still exist in both corded and cordless, light duty and heavy duty tools.