Okay, the second bike appears to be a much better bike than the first, I'm not that familiar with that brand either, but downtube shifters, better components, better frame materials. I think that would be a better buy. However...
Preface: Take this with a grain of salt.
Might I suggest a vintage mountain bike? For a 1-2 day a week ride down a bike trail (unless your gf has a road bike and rides pretty hard), plus grocery runs in a city, I would lean towards a vintage MTB with road-esque tires. The tires are cheap enough, and the MTB will be well suited to having a rack on the back, going off a few curbs, and getting put in a bike rack.
http://toronto.kijiji.ca/c-ViewAdLar...7&ImageIndex=6