Bob is not, strictly speaking a "short neat haircut". Firstly, a bob is a style not a cut as you have to actually put your hair into a bob. But secondly, you need to have longish hair in the first place to be able to put it into a bob.
Unfortunately, though, I can't actually think of anything better...
Actually a Bob is described as a short cut for women, popularised in the 1920s. It's not short as in a crew cut but it can be fairly short hanging across the cheeks.