Tough call. I'd say Bethesda's because in Fallout 3 and 4 (I'm not sure about the other ones because I didn't play much of them) their power is clear and showcased to all of the wasteland. But Obsidian's take is the Brotherhood of Steel as bunker hermits. Bethesda's in general, in my opinion, fits them more, since they are very powerful and technologically advanced, and prove it through their presence in both FO3 and 4.
To sum it up, Bethesda's.