iTunes: Why do some internet radio streams show up in my main music list and some do not?
I tried both streams like you did and at first I received the same results you did. BBC Radio 6 showed up in the music but The Current didn't appear to.

Then, I tried typing "The Current" into the iTunes search box while Music was selected in the sidebar. The result: The Current from Minnesota Public Radio pops up as the first and only search result in the iTunes main menu. 

What I believe it that because this radio listing contains a blank album and artist field, iTunes gets a bit confused but ends up placing them "somewhere" amongst your other entries with blank album and/or artists fields. It can appear to be lost there because there doesn't appear to be any particular order those entries are sorted by-default. (I am almost certain you can do something about that if you really wanted.)

IF you are still interested in seeing your music with those streams, I would recommend just creating a smart playlist that excludes them. Try this a the rule to use in the smart playlist (Cmd-Opt-N to bring up its dialog, btw):

"(Kind) (is not) (Internet audio stream)"



Hope this was helpful.

-b