That website's creator puts an obviously-designed sand-sculpture in opposition to a putatively undesigned sandstone formation. But once one grants the possibility of an unspecified designer, I don't think that there's any way one can attribute randomness or lack of design to the sandstone formation, or to anything else. There is no counterexample that couldn't be the work of an omnipotent and hyper-involved designer. Sand-sculpting winds, though appearing random to us, might in fact be as directed as an artist's brush.
Back to biology: Let's say in twenty years we can successfully design an microorganism. (Supposedly this has been done already, using other species' spare parts.) We hold up an obviously-artifical organism to an allegedly undesigned one. What's the difference between the two?
Considering both organisms must follow basic laws of chemistry and biology in order to thrive, I don't think any meaningful difference can be specified.
The Christians among the ID movement want to focus on specific examples of order to leave some sort of space for God when they should be concentrating on the overall nature of that order itself. Ontology and metaphysics don't have the cultural cachet of natural science, but they really are superior modes of inquiry for topics such as these.