My bad, but I think you've somewhat answered your own question here. In this case saurischians and ornithicians are being used as phylogenetic terms to represent groups of evolutionarily related animals based on a key morphological trait that was easily distinguishable by comparison to birds versus lizards. The fact that bird and bird-hipped dinosaurs have similar pelvic structure is likely a simple case of convergent evolution.

The terms have to do with whether the legs are positioned underneath the body (like a bird), or splayed out to the side (like a lizard). I don't think you're correct saying birds are saurischians, apologies if wrong.