It is a unique arrangement and there are arguments for saying, "you're both right." They are Newtown Juniors playing in the Souths Juniors competition in the sense that the club, through representatives, run and fund the club.
But unlike the Easts Juniors that play in the Souths Juniors competition, Souths Juniors contributes to the funding of the Newtown JRFLC, so they have claims to calling them, "Souths Juniors," and should the ocassion arise, Souths would be able to pick players from Newtown JRLFC to play in Souths Junior Rep squads. However, Newtown were able to secure a special arrangement that will see the Newtown RLFC claim a development fee should a player come good out of the club (maybe brendo, maybe someone else).
It's been said on this forum before that Newtown would ideally have funds to claim a district of its own, as it used to. But the funds aren't there so this is the best arrangement possible.