I believe cloud by definition means third party, but something like the host wont know what it is storing, just who accesses it. So in OP's case if his computer is shutdown, the server wont be able to serve the file, but if it is on "cloud" there is a lot less chance of downtime and greater latency as "cloud" can have multiple data centers around the globe to server same file.
Since as long as I remember and probably even before that, when explaining networking to executives, marketing and other technically challenged people, we use drawings. In these drawings the part representing the internet is for some reason drawn as a cloud, probably to convey that some magic we don't want to get into the deatils happens there.
Now you've got it, the cloud, a.k.a. the internet symbol for ignorant people.
