Last time I researched this (when I settled on a desktop condenser mic), the only headphones that had good mic quality were the very expensive ones, $200 and higher and/or the professional ones (the ones you see used in sports on the field).

The problem is that, just like cameras, size matters for mics. In order to even sense the full frequency range of the human voice it helps to have a larger diaphragm size. For headsets they range from microscopic to tiny which results in the voice recorded with the headset sounding like "telephone" voice (ie it's designed to capture the most important frequency range of the human voice, the medium frequencies) however, to me that sounds bad, artificial, cheap so I wanted more. Capturing more of the frequency range allows you to sound more natural, you know, like you are actually there in the same room talking to that person.

Depending on your usecase this may or may not be a consideration.

EDIT: as for noise issues, it's always best to avoid recording it if possible but even then there are software techniques to remove it post-facto. In order to achieve the former you can get super/hyper-cardioid pattern microphones that are very sensible recording sound that comes directly in front but not from other directions. However, no pattern will save you if you use a condenser microphone in a noisy room. It may be possible to use a second mic positioned away from your mouth and use software to remove the noise by essentially computing the difference between the two (some headsets can do that). Alternatively you can go for a desktop dynamic mic, just make sure you speak loud enough, but at least you won't get noise :)

> I always thought a headset is better than external mike…

It is, because users don't have to worry about mic technique or having a studio-quality room.

The only concern I generally have with microphones for the vast majority of my life is music. Only recently have I concerned myself with the sound quality of online meetings. :) Meaning I'm not the best to ask, but maybe better than anyone else you know. But again, I think anything that's wired with microphone that doesn't have to deal with what comes out of speakers (feedback) is going to be a big improvement, even headsets. And moving around is a factor, as I'm leashed to my desk (though with a long enough headphone cable extension that I can work the resistance bands during long meetings). If I were worried about background noise, I could switch to a dynamic microphone.

Anyway, if you've got a headset you're happy with and no one complains, I'd just use that if it were me. Again, I've got the "good stuff" lying around already, so I just use that. I would not recommend anyone else go that road unless they have other uses for the gear.