How does the game change when you get introduced to a girl through mutual acquaintances? She is actually friends with my (now former) oneitus, but we are in different social circles (our group of friends don't hang out). Asking a stranger out is easy because you don't have to worry about the outcome, but in this situation since we all kinda know each other, how do I play it? Do I try to get myself into her social circle? I need to build attraction and comfort before asking her out on a date, but how do I do that if we are in different social circles? Or should I not give a fuck and just be direct and say you're cute we should grab some drinks together?

Pretty much this. Introduction through friends is awesome social proof, my best relationships always started because of knowing mutual friends. It also gives you something to talk about. If she says no, it's no big deal, go back to building attraction/rapport/qualify when you're in social situations with her. Then try again later.

If I went too direct and got rejected, everyone in that social group would find out and pretty much ruin my chances with the rest of them, no?

I was introduced to this girl through friends, went direct and approached her later to exchange numbers and make plans. When I texted a couple days later to invite on said plans, I got no response (rejection).

It hasn't affected my standings with the rest of the group. I think you're worrying too much, sir. Of course your other option is just wait until you see her in the group setting, then game her from there. Do you have her number already? How often do you see her through friends? How's your interaction with her so far?

I've only talked to her once and I rarely ever see her. I don't have her number and we are in two different social groups. The only way I can really see her is if I tell my oneitus to hook me up with her....but that'd be weird.