I have heard a couple different opinions regarding the non-disconnecting 'cuts' in the standard definition (first in the linked wiki) of genus. I am of the opinion that we require a closed cut to have the same initial and end points, but I know of some who would allow an infinitely straight cut along an infinite cylinder, for example.
But the shape in your second link, I don't think, is infinite in each direction. Thus it's really just a cylinder with a little knob. In other words, it's a pair of pants. So it has genus 0, like a sphere, as I can't cut any circle in a pair of pants without some fabric falling off.