It might be that (being a puppy) Dexter is so excited to play that he forgets to do his business when he's outside, and then can't hold it when he comes back in.
I would suggest regimented bathroom times (after meals, sleeps and plays is usually a good measure) where you take him out alone to do his business so that he cannot be distracted by the other pup.
"We used to praise him and give him doggie treats every time he went outside, but he loves the outside so much, he'd go out, play and come back and expect a treat."
I'm sorry if I'm misinterpreting (curse you internets!) but it sounds like you may be rewarding too late to capture the right behavior. If your dog goes outside, does his business, and gets a treat for coming back in he will associate coming in the house with the reward - not going to the bathroom outside. In order for him to understand that you want him to eliminate outside you have to be right beside him to reward within seconds of him finishing.
Just take the house training back a few steps and you should be fine
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