Coughlin had to back down to not only get through to the guys, but to keep the team on his side.
Bellicheat isn't as militant as he is just weird. What helps him out, in addition to his resume, is that the vets on the team have bought in and set the tone.
Of course coughlin toned down, he's wouldn't be where he is today if he didn't but there's no way in hell you still can't label him as a disciplinarian. And same with belicheat.. There's reason these coaches are succesful but that doesn't mean they're not "disciplinarians" type coaches.