For me it's not as simple at bass vs clarity. I love bass, but I hate muddy bass.
Since typing about "creamy thumpy boomy tight quick slow" sounds never ever really gets the point across, lets put it this way:
Bad bass = Car-load of teenage boys behind you at the drive-thru at Taco Bell cranking their subwoofer so loud that the bodykit they glued onto their Civic is starting to fall off. It's physical bass rather than audible bass and just distorts every other sound next to it. It's just unpleasant. This is the bass you feel up your butt basically, and some people like that, even in headphones. The same way some people get off on treble so energetic that it's physically painful. They like one type of sound and chase the most extreme sensation of that sound.
Good bass = The THX sound when you watch the original Star Wars movies or listen to early '80s electronica where they experimented with a lot of analog synth bass. Really deep, but it stays in its own space and lets the mids and treble do their own thing, almost like the audio is coming through in 3 or 4 separated slabs. This type of bass can coexist with clarity. This is the bass you feel in your teeth and chest rather than your butt, the exhilarating kind. Because it stays separate from the other frequencies, it can even harmonize with them and create a constant "power chord" effect.
Unfortunately finding the good type of bass is such a fragile and subtle endeavor that a lot of speakers don't nail it and fall off to the needle point, either resulting in boomy/muddy or shrill/piercing.