Probably the best as far as exercise is whatever gets you to ride it more often or further
. A bike with no gears is going to be fun, but with gears, it makes it easier to ride esp if you have a hill (and you wait....you THINK there's no hills.....when you're on a bike, you'll wonder where that dang incline IS b/c all of a sudden, you're having a harder time pedaling and you're going slower and slower, LOL).
I think for just casual, bebopping around (I mean, if you're talking about taking a dog on the bike with you, you're probably not going to be going for a 40 mile ride and you probably don't want to be going 28 mph) the bike she's looking at is fine; I say a mountain bike b/c I personally would rather have one that's more versatile if I've only got one bike(like what if you find that you really LOVE riding a bike and want to go further or faster sometimes when you're by yourself???) and also, the tires are a lot less prone to flats, and they have more traction than slick, skinny tires.
If you're looking for exercise, you want to get the one that you really enjoy riding.
One thing you can do is go to a bike store (a real bike store that sells only bikes and bike equipment) and test ride a few. Most bike stores are happy to have you do that...that way you can see a different range of types of bikes and what feels best to you. It's no good to get a bike and then have it sit in the garage (or wherever you keep your bikes).