Internal testing is Alpha, beta testing should be done by somebody outside. But yes then or just later when they have a timescale for the required fixes would be good.
This is not true, and varies alot between companies. Take Star Citizen as one example... but... I also have seen companies doing closed Alphas and Betas and open Alphas and Betas... The list can go on for quite a bit. Elite Dangerous itself had a Gamma back in the day for the kickstarters.