It's not "teleportation" in the Star Trek sense. They need to get the entangled photons to both places so observers at both sites can see them. That means that nothing can interact with the photons on their path (or rather that any interaction needs to be precisely defined such that it doesn't break the superposition). So you can't just send them through the air.

That confused me as well. They seem to be talking about the relocation of one of the entangled pairs to its intended position. But it doesn't seem like what I would call teleportation, since it seems to be moving one of the entangled pairs along a path. They do describe one thought of teleportation in the beginning that fits my thought; copy object data, destroy object, send object data to second position, create object. But what the Chinese did doesn't seem like that unless I too am missing something.

It is the information that is (in a sense) being teleported. The photons are the transporter rooms not the people being beamed aboard (to make a tortured, lossy analogy).