This is a topic that I have tried hard to resolve for myself, and am not quite there. But here's how I currently see it. I won't be offended if anyone gets 1/3rd the way through this and stops reading. I'm sure it comes across like I'm pontificating....
Imagine an umbrella, a bunch of rods all meeting at the top.
Each rod is a different style of music. Blues, rock, country, jazz, dance, rap, metal, etc.
High up on the rod (the top, where the rods meet) is how "mainstream" the song is. In the middle, it's "standard" (for that genre). The furthest away from the top at the far end of the rod, it's "alternative" (for that genre).
A song can be directly on the rod (a pure 'rock' song), or between two rods. Country-rock, rap-metal, punk-rock, etc.
Now we test it out..... Let's start with a few "pure" (on the rod) songs:
Bryan Adams "Summer Of 69" = mainstream rock
Ted Nugent "Cat Scratch Fever" = standard rock
Breeders "Cannonball" = alt rock
Metallica "Enter Sandman" = mainstream metal
Iron Maiden "Run To The Hills" = standard metal
Faith No More "Caffeine" = alt metal
Will Smith "Getting Jiggy With It" = mainstream rap
Run DMC "Run's House" = standard rap
Watsky "IDGAF" = alt rap
Sometimes a song is a "fluke" ... it starts out as an underground song but finds pop success, so it gets both labels
Eddy Grant "Electric Avenue" = mainstream alt dance (with new wave influence)
So with the songs being discussed in this thread, here's how I would classify them (with other examples thrown in):
Radiohead "Creep" = mainstream alt rock
Radiohead "I Can't" = alt rock
Red Hot Chili Peppers "Under The Bridge" = mainstream alt rock ballad
Red Hot Chili Peppers "Nobody Weird Like Me" = alt rock
Soundgarden "Black Hole Sun" = mainstream alt rock
Soundgarden "Rusty Cage" = alt rock
Counting Crows "Mr Jones" = mainstream rock (I don't think they are alt, they were a nice alternative TO alternative music)
Counting Crows "Angels Of The Silences" = standard rock
Pearl Jam "Jeremy" = mainstream alt rock (this was originally alt, but became the mainstream - killing what was mainstream before it)
Pearl Jam "Spin The Black Circle" = alt rock
Smashing Pumpkins "Today" = mainstream alt rock
Smashing Pumpkins "1979" = mainstream rock (the only slight bit of alt to this song is that they are an alt band at heart performing a pop song)
Goo Goo Dolls "Slide" = mainstream rock (nothing alt-y about it)
Goo Goo Dolls "Name" = mainstream rock ballad
REM "Losing My Religion" = mainstream alt rock
REM "Pop Song 89" = alt rock
REM "Shiny Happy People" = mainstream rock (like "1979" it's an alt band at heart doing a pop song, but the song itself is not alt)
Anyway, that's my take on it all. I don't think it's totally workable. But it's the best I got.