If that's the case, then Apple TV isn't streaming either, and Apple calls that streaming.
arn
I don't have an Apple TV, so I'm not positive about how that works. I thought that on an Apple TV, rental videos start playing nearly immediately (after a short buffer builds up based on available bandwidth), while rentals on other iOS devices have to download/buffer the entire video before starting playing. So, that could actually be streaming on an Apple TV.
Again, the difference is minor and with smaller music files might almost be negligible to users.
True streaming: the file is broken in to lots of smaller pieces and starts downloading at the initial point requested by the user. If this happens to be the beginning of the song, then it would almost be identical to the progressive download streaming below. However, in this scenario, the user could skip near the end of the song before the whole thing has been downloaded. If the stream download has not reached the point the user wants to hear, this normally clears out the cache of the portion of the song that's been downloaded and basically starts the stream over from scratch at the new point. As far as I know, most streaming protocols don't store any portion before the requested start point and usually ditch their cache as soon as possible.
Progressive download "streaming": the entire file is downloaded from start to finish. Playback can begin as soon as the required buffer is filled. You cannot skip/scrub past the farthest point that the file has been downloaded = you have to wait for the file to finish downloading before skipping to the end of a song.
Apple is using progressive download "streaming" and storing the file in a cache. Based on the video, it doesn't appear that the cache holds onto the file for much longer than app has interest in it. In the end, this is just a semantics argument, and to the user, Apple's solution feels nearly exactly the same as true streaming. Still, the 2 solutions use different protocols on the back end and Apple's is not streaming.