Unless you and your girlfriend's girlfriend go out without your girlfriend, it might be simpler for your girlfriend to do the introducing.
You didn't name her, so I'll make one up. I'm fond of the "This is Betty" approach. Why the need to label her role? She's Betty first, and her relationship role just reflects who she dates, not who she is. Depending on the context of the introduction, it will be obvious that she's "a friend or something" and that's all anyone really needs to know. If they're curious, tell them to ask your girlfriend for details. They're her beans to spill.
If you figure you've got issues to deal with before you're ready for multiple-dating, then you're probably right. But that won't take forever, so some advice for when you're ready:
First, talk to your girlfriend to see how she would feel about you dating Betty. Not to say she should have the right to veto your relationship, but it's good to know where she stands. But at the end of the day, your girlfriend doesn't own Betty and she doesn't own you. She doesn't have the right to tell either one of you whether you're "allowed" to date each other.
At the same time, ask her if you're crazy for thinking Betty is sending you mixed signals. Maybe that's just how she is with everyone; some people are like that.
“As I am sure any cat owner will be able to tell you,
someone else putting you in a box is entirely different
from getting into a box yourself.” —bisexualbaker