What's the difference between 打ち合わせ and 会議 and ミーティング?
To me it seems that 会議 may be more formal, but I'm not entirely sure.
打合せ is more generic. When you go abroad to meet your boss, your colleagues or customers, you'd say 打合せ.
I use ミーティング for when I need to discuss a point with someone. If you're a student and discuss with you advisor once a week, that's it.
会議 is defined as "there's a boring speaker, and everyone around is sleeping" :) That's in fact a conference (even a small one with a handful of people).
For a business meeting: 会議
It basically means: "To meet for deliberations"
It's equivalent to the gairaigo: ミーティング (which I don't like much because there's a widely used Japanese term already)
打ち合わせ is used for meetings of people ”with a common interest". Meeting of a club maybe? As the link explains, it has it origins in Gagaku, where musical instruments need to "hit (taiko) as a unit and fit with the music"
From my experience:
It might be specific to my company.