I agree with both the OPs thought. Too loud and too busy. However, both videos are horrible examples.
The first guy is using soooooo much arm and core muscle energy to hold back. And, while he talks about being solid and keeping perfect time loud or quiet (and is using a click!), his time and feel is all over the place within the non-existent pocket. His technique is dreadful. He should take a lesson on how to hold drumsticks.
The second guy clearly tries to use metal drumming backing The Beatles to demonstrate simple vs busy playing. We all get the point. However, the example is totally extreme. I'd like him in a Beatles style rock band as much as I would like Ringo in a metal band. Wrong guy, wrong job. Both styles are valid within their own contexts.
Anyway, to reiterate...
1: Have the ability to play softer.
2: Be able to reduce or expand complexity to fit the piece.
I'll add...
3: The click is a tool, not a sign of weakness. ALL MUSICIANS (not just the drummer) can keep better time with practice against a solid reference.
4: Know where the 1 is, always. If the fill is going to throw you off, don't play it. Ask your instructor for help before the next rehearsal.
5: More than anyone else in the band, lock in with the bass instrument. Drums and bass can get establish groove without the need for any other instrument.
However...
6: Play the set of drums you are comfortable with. If it's large, that's great. If it's a four piece, that's great too. The drummer doesn't get to tell the pianist to play a 49 mini-key synth and leave the big ol' 88 weighted key digital piano at home. (OMG - Who needs 88 keys?!?!) If the bassist with 5+ strings gets after you about the size of your kit, remind them that Jaco only needed four. And Jaco's bass was clearly the best bass for metal, too. And frets are just speed bumps. Yes - #6 is clearly meant to be funny. But seriously - don't tell the drummer what kind of kit to play.