Yes, in an equipped meet, one would wear briefs AND a squat suit. I'm not sure where your confusion is on this aspect. I have no idea why someone would choose just briefs, outside of financial considerations. There may be circumstances in Strongman where just briefs might be used. I've seen events where briefs were allowed but not full suits.
For training purposes, an equipped lifter may wear just briefs sometimes just because they don't want to put on the full suit.
As has been mentioned, wearing briefs is NOT the way to go if you want your hips to get stronger. That's backward thinking. The overload would be on the knee joint, not the hip joint.
Bottom line is, if you're a raw competitor, training in briefs is probably not a good idea, ever. Unless you compete in strongman as well. Or for some reason you just like wearing briefs...