You're quite young. I wouldn't worry too much about your situation. If he initiated the therapy (and the apologies), that's a good sign. But it's not necessarily a bad sign if it came from you. He's agreed to therapy and it sounds like he's genuine in his expressed desire to be with you.
I would be highly sceptical of the "sex addict" label and the value of therapy at his age. Especially at 20, the sexual drive is as powerful as it will ever be for a man. If he were 40 say and you'd been married 15 yrs then you'd have more to worry about. So, the short answer is, in your case, once a cheater doesn't have to mean always a cheater. Talk it through with him and in future, don't let problems fester because things that don't get discussed can be the pebble in the shoe of a relationship. Good luck, don't sweat it.