Are runway safety areas for both emergency landings and aborted take-offs?
Yes, the runway safety area is to provide a buffer zone for aircraft that leave the runway for whatever reason, including rejecting a take-off late or by overshooting on landing. It's not specifically for an "emergency landing": sometimes the emergency might only arise during the course of the landing, for example tyres bursting or pilot error.

If it were only for emergency landings, then parallel runways probably wouldn't need to all have a RSA, because the distress aircraft could be offered the one runway that has one. But in fact every runway needs an RSA, because any landing might suffer a runway excursion.