Well, I am sorry to say, that I have found the solution to my problem and it is so embarrassing, that I almost didn't want to post it. At some point, I must have selected "Show Music Available Offline", possibly before I was getting on the subway, so I could see what music I had downloaded onto my phone and could listen to it underground, and forgot to turn the setting off.
As you know, when you turn "On" "Show Music Available Offline", that is really the same thing as "Show Only Music that has been downloaded onto my phone". Any music that you "Add" to "My Music" but without actually "making it available for offline use" (meaning - download it on to your phone), will NOT show up when this setting is selected.
As soon as I turned it off, all of the music I had been adding to My Music from my desktop, ipad etc, all appeared in My Music. In other words, all the times that I was clicking on the "Add Music" button, it was being added to My Music, I just couldn't "see" it because it was set to only show Offline Music.
Thx