Should the bit in the impact driver have free play

The main difference to pay attention to is a drill vs. a driver.

At the core, they are essentially the same in that they turn a bit.

A driver, however, also has a clutch. That's the key difference. With a drill, you want continuous power on the drill bit. However, with a driver, you want continuous power up to a point at which point you want to stop the power to avoid stripping the screw, over-driving or breaking the screw.

You can drive screws with a drill, you just have to be more careful to avoid the problems listed. And while you could drill a hole with a driver, I wouldn't advise it, as the clutch would make it a pain as it would slip as you go.

Ideally, you have both...for no other reason than you can keep the drill bit in the drill, and the driver bit in the driver, and not have to constantly swap bits as you drill pilot holes and add screws right behind it.

As for a "regular" driver vs. an impact driver, you use the latter when you need high-torque. A typical example would be construction screws where you may be screwing through 4x4 timber and the like.

Is this the first time you've used an impact driver? They sound and feel very different. The clicking / banging you hear is not the clutch, it is the hammer mechanism striking inside the tool. (Does your impact driver even have an adjustable clutch? It is not common on impact tools.)