A tiny robot with moves inspired by caterpillars and jellyfish

Look at my crumpled paper robot! It goes wherever I direct it with puffs of air!

Well, I think the argument is that you're looking at a tiny piece of a larger equipment, which contains the power source and needs to manipulate this smaller device using magnetic fields. So, probably the entire set of equipment would be the robot.

While still very cool, I think the way it's presented here implies that the little rubber part is autonomous in terms of power, muscles / actuation, sensing and local planning capability, which is just not the case.

That's actually a really complicated question because 'robot' is a poorly defined category. You run into similar problems trying to figure out a definition for "what is life?" that includes everything that we think is obviously alive (like worker bees and frozen-but-revivable organisms) but doesn't also include things that we think are obviously not alive (like fire).