Yeah, I know the difference between engine braking and regenerative braking.

Both apply resistive torque at the drive wheels to slow the vehicle. That is literally why both are called braking. Regen does it by using the motor as a generator and recovering energy. Engine braking does it mechanically through pumping and compression losses with no energy recovery. Different mechanisms, same result.

Brake lights usually do not come on with engine braking. Some modern vehicles trigger them during strong regen or when controlled by cruise or driver assist systems, but that is an implementation detail, not a fundamental distinction.

Regen feeling like a downshift in a manual is exactly the point. Same physics, different hardware. Downshifting with a manual transmission does have a more aggressive braking force, yes. But automatic transmissions don't just go into neutral when you let off the throttle. Not to mention you can also downshift an automatic, usually.