In my experience, places where engine braking is prohibited tend to be with the intent to reduce engine braking by tractor trailers or other large apparatuses, which can be very loud. In much of the U.S, posted signage prohibits "Jake braking" which specifically targets tractor trailers and other large trucks. [0]

[0]: https://travel.stackexchange.com/questions/96447/why-prohibi...

I will chime into that, though, that they are prohibiting large trucks from doing it. They don't care about cars doing it. (The wikipedia page explains the difference and why some places prohibit trucks from doing it.)

Edit: That said, I fully agree that most times I hear folks talk of engine braking, they mean shifting down to slow somewhat faster than just coasting.

Engine braking is prohibited due noise from large diesels using compression brakes aka "jake brakes" - specialised engine braking hardware. They make engine braking much more effective, so that trucks can slow down more without heating the regular brakes as much, but they can be loud.

The parent is essentially right; engine braking is using the engine to slow the vehicle down.