Auto-braking when turning into a slippery corner?

Yes, the brakes are being applied.

When you push the brakes harder, you are probably activating the "Hill Start" function. The Prius tells you that this is on by beeping and turning on the traction control light (in a situation where it could not be traction control). Using the same light for both functions was not a great idea as it has confused an awful lot of people.

If a car just has a standard differential and no traction control, then when either drive wheel completely loses traction and spins freely, the differential also spins (slips), no power is delivered to the other wheel, and motive power is lost.

"Traction control" works by sensing when a drive wheel spins and braking that wheel enough that it no longer spins freely and some power continues to be delivered.

Another way to avoid loss of traction when one wheel slips is to provide a limited slip differential.

 

This is the VSC activation to keep your car from sliding sideways by applying the brakes to each wheel independently.