A great many pop tunes are like the Stevie Wonder tune you reference, they float back and forth from the Major I to the relative minor. Since these keys have the same signature, it can be very difficult to tell what key they are in. Here in Nashville, where the number system reigns, you have this problem all the time.
To me, you have to look for the dominate chord. Is there a strong sense of Bb7 resolving to Eb, or is it G7, resolving to the C minor. Also, I would pay attention to the last chord. In music that is Modal, and not tonal, that doesn't utilize dominant chords, as is the case with much r and b, than the last chord can be used to determine the key. I don't know the Stevie tune discussed here, but if it seems to wander between the Eb and Cminor, which ever one it ends on can be used to interpret the song.
In the case of the Stevie song, it actually doesn't matter what key you call it, or what numbers you assign the chords, since the chords are the same in Cminor and Eb major (accept the dominant V chord) The only difference would be if you are the session leader handing out charts to play. As long as the charts matched up so the musicians weren't confused, you should be OK.
If this is just for your own study, it truly won't matter if you interpret the song as being in C minor or Eb.
It may be helpful to chart it out in both Cminor, and then again in Eb so you have two interpretations. Then you can use which ever one you feel comfortable with. I am careful to say interpretations here, because I don't think there is a "correct" way in this example. Again, the prescence of the dominant V-I or ii V I progression will let you know.
Also, to try to answer your other question, Key usually equals key signature. No sharps or flats usually means C major or a minor (in a tonal music setting) The exception to this is in modal music, where the same key signature could indicate d-dorian (such as "So what") or G mixolydian. Also, a lot of jazz won't really have a "key" but will use accidentals and cascades of ii-V-I progressions, to suggest different keys at different times. What key is "Giant Steps" for example? A lot of modern music is basically chromatic, and a lack of a key signature doesn't mean C major by default.
