No problem. We get confused on a regular basis! đ
Your belief is correct. However, I expect that everything will work even easier if you use iCloud. Fortunately, you can experiment with it and, if you don't like it, turn it off. (See below.)
In the absence of iCloud, you purchase music on one iPad and, because of syncing, the computer's iTunes library has that music. However, the other iPad and the other computer do not have the music until you manually download it (at no charge because you're using the same ID). With iCloud, everything will be automatically distributed to both iPads and both computers and you can then manually delete from the iPads as desired. BE CAUTIOUS about deleting from the computer's iTunes library as there is only one iCloud backup file.
Computer iTunes info tab: