Rockets that use electrical pumps for the fuel and oxygen pumps. Usually rocket engines use some of the fuel and oxygen to run these pumps, which makes them a lot more complicated.

https://en.wikipedia.org/wiki/Electron_(rocket) Is the big player in the field afaik.

Magic? Non-satirical response, there's companies like Rocket Lab that make electrically powered turbopumps to simplify their rocket engines but that comes with very steep power requirements. The Electron rocket literally has lithium ion packs that drop away from the rocket as they are depleted and the weight of those packs really starts to add up because those turbopumps use absolutely insane amounts of power to keep up with the pressure and fuel flow requirements of the engine.

Not counting turbopumps for chemical engines there is also an electric propulsion when in space.

To expand on this there's these things called ion thrusters that use an electrically generated magnetic field to accelerate gas particles at extremely high velocities out the spacecraft thereby generating thrust. These are stupidly efficient but also very low thrust. Due to the high power usage I don't see much of an application for even incredibly dense batteries as you still need a sufficiently large power source anyway - though I'm no rocket scientist.

Presumably a rocket with a power source separated from its exhaust, and that flings its exhaust material out with an electrically-powered device.

So basically think of a 'reverse railgun'.

I was thinking of https://en.wikipedia.org/wiki/Electron_(rocket) Which uses electric power to pump the fuel and oxygen into the nozzle.

That's an ion thruster. They've been around for a while.

Every railgun is a thruster - or gun for that matter: https://what-if.xkcd.com/21/