Program files have to be installed on the system drive on each machine.
Ideally your library files would be at E:\iTunes given the external drive shows up as E: and the media folder would be at E:\iTunes\iTunes Media.
From the point of view of portability having the media folder inside the library folder is the key to making sure that when you move the library between systems everything keeps on working properly. In the right layout it wouldn't matter if one system wanted to shift the drive to F:, for example, you could just choose the library file at F:\iTunes\iTunes Library.itl and everything would work as normal.
That said, provided both systems assign the same drive letter to the external drive you should be able to use the same library on each even with a split library/media folder arrangement.
Since you say the two machines currently have different playlists I think you may be mistaken if you think both are using the library files at E:\iTunes. To check which folder holds the active library on each machine you can use my script iTunesXMLPath.
The post on should explain the steps to get from the current arrangement to a fully portable library, if you want/need to. Once we've established exactly where things are I can write our a set of specific steps for your circumstances rather the general overview given in that post.
Message was edited by: turingtest2