I'd also never heard of "the bird" (other than that it's the word...) before, but from those images I can't see how it would be considered the inverse to the push-up.
Push-ups primarily target the chest and triceps, whereas chin-ups/pull-ups primarily target the back and biceps, so they're a strong contender for the inverse exercise to the push-up. Lego Stormtrooper's suggestion of the Inverted Row is also good.
Between the two, I'm not sure which would be the better option. It depends what equipment you have access to and what kind of shape you're in. If you have a chin-up bar already, but you can't adjust (lower) it to allow you to do inverted rows, then that makes the decision for you. However, I've always found inverted rows a fair amount easier than chin-ups or pull-ups, so if you're starting out I'd suggest those first - you can always progress to chin-ups and pull-ups later.