As many people have said, don't show her your games on a first date. It's too risky because
(1) you don't know if she actually is a "gamer" (I know lots of people who played Mario back in the day and a bit candy crush or clash of clans now that wouldn't be caught dead with the gamer label);
(2) even if she is a gamer and willing to game on the first date, you don't know her tastes well enough yet to pick a game to show off how far the medium has come. For you it might be something like the last of us, for her journey or even 3d world. Hell for her candy crush might be that game! Choose wrong and you might just end up boring her to tears or worse condescending her in the nerdiest way imaginable;
(3) lastly, and most importantly, even if she is a gamer, that still doesn't mean she wants to see your games or particularly cares "how far the medium's come." In fact, if this girl is into you and hopes you're into her in more than just a platonic way, busting out a video game on the first date might send the wrong signal.
All that's not to say that you should hide who you are. I've always been up front about my interests with women. But I usually wait until they express an interest ("Wow, is this the new Mario Kart? I used to play the one on 64 growing up!") or they've demonstrated to me that they're the kind of person who's really willing to give them a shot before inviting them to play with me. Oh, which reminds me:
(4) if and when the time comes, please don't sit your date down and play a game like the last of us in front of her. It's not even just the violence aspect of it. There's nothing romantic about sitting there watching someone you don't really know play a video game. It's not like a movie where you're both sharing the exact same experience. She might feel awkward or unable to talk to you while you do battle with fungus-zombies.
Date experiences should be shared experiences. For example, my girl and I play shovel knight together, taking turns, laughing our asses when we die, and talking the whole time. It's awesome =) But I sure as shit didn't lead with that when I met her!
Good luck on your date op!