Are all bad habits worth spending time fixing?

I don't think there's anything wrong with fingering a chord in your own unique way. If for some reason you had to finger it the "correct" way, it would just take a couple practice sessions to get used to it. When I was working through John's jazz book, I'd change fingerings for chords based on what came before and after it - if you can keep a couple of fingers on the same strings (even while moving to a different fret) and just re-position the other two, it made chord changes much simpler, as opposed to using the "correct" fingering for the first chord and the "correct" fingering for the next one, which might require you to re-position all 4 fingers.

My new teacher is trying to make me more critical and self-conscious about how chords ring out, which strings are properly muted, which strings are unintentionally muted, etc.

In the process, which is ongoing, I have to unlearn some old chords fingerings. So perhaps be self-critical and try to analyze if you're able to form a solid fingering of the chord on time, every time, and if every note of the chord rings out as you want to. And otherwise be open to learn a different fingering, which can of course be frustrating.

This is something my teacher is very particular about, and I see his point, because he has an amazing rock-blues style of playing with lots of percussion and muted strings, and it sounds like crap when I try it, because I haven't developed good muting technique, and I also use some fingerings that tend to sound muffled or buzzed.

I tend to look at it from a perspective of context/voice leading. In other words, it depends on what the fingers are doing before and after the chord in question. There are several chords I can think of offhand that I finger in multiple ways depending on what's going on around it. That said, I often encounter fingering suggestions I can't make work well. I don't see any harm (unless it is painful in a bad way) to continuing to work on fingerings I struggle with, something I base on a presumption that there's a reason players often use the fingering in question.

I tried fooling around with the chord you illustrated. I can make it work either way but the first fingering you show is much more comfortable. I made a new for me exercise out of switching the positions of the first two fingers leaving 3 and 4 in place. I wouldn't say playing it the second way is a 'bad habit' per se. The caveat is I'm not much of a jazz guy and maybe there's much more overall utility to the first fingering which is why you encounter it more.