Over on the linked question, I talked about how engine braking works to reduce fuel consumption at a high level:
Coasting: nothing much. The transmission is effectively disengaged (it's more complicated than that but it's a reasonable approximation). The engine is idling - burning fuel to keep itself spinning.
Engine braking: the transmission is engaged and the whole system's net friction (from the wheels, axles, driveshafts, all the way to the moving pistons) is acting as a brake. The engine is running in a vacuum state (throttle is effectively closed) and the motion of the wheels are helping to keep the engine spinning.
Do either cause undue wear? No. I wouldn't advise engine braking down the hill in a low gear near the redline (as you might run past the rev limiter) but you aren't anywhere near that state.
Does one save more fuel than another? Yes, engine braking is much more efficient. I talk about why that is in this similar question but suffice to say that the free energy provided to the engine from the turning wheels helps a lot. Put your car in the highest gear available to minimize the drivetrain friction on the way down the hill.
There is a nice video on Engineering Explained that discusses the whys and hows of engine braking and efficiency:
The tl;dr is:
- Proof by existence: you have a numerical measurement that shows that fuel consumption is less when using engine braking.
- Passes BS detector: if the wheels can keep the engine turning, why should it use fuel?