How would you introduce your family to a girl you're seeing like this?

[Rob]

This girl and I started getting close over 2 months ago. She had been in a relationship that had lasted over a year and a half that just ended with a horrible break up a few months before we started getting close. We'd seen each other only rarely before then, but after helping her out of a rough spot we've been getting closer every week. She invited me over to her place every weekend, and things eventually progressed from hugs to making out to eventually sleeping with each other. After that we agreed we had to talk about things.

I asked if she wanted to be exclusive, she said she likes me and wants to keep dating but can't handle being 'tied down' right now. It's not that she's sleeping around or anything but flirting with people at the bar with her roommates, but she just doesn't want to hurt me if she makes a mistake some night when I'm not around. Eventually we agreed to just go with it... and since then we've been seeing each other more often.

Heck, she's coming to my place for a fake thanksgiving meal after work. Her roommates have asked me why we aren't official yet (to which they said 'that's bs she only ever talks about you' when I told them she didn't want to be official), we're seeing each other 2-3 times a week and text each other all the time even when we can't be together.

Am I just getting too caught up in trying to label this? This is easily the best relationship I've had with any girl, and yet this isn't even an actual relationship! What should I even call her to introduce her to my family this weekend? 'The girl I'm seeing?'