+1 on the Eagles for harmony. Their surround-sound version of "Seven Bridges Road" that came on the Hell Freezes Over DVD is amazing. The Eagles bring a super-tight 3-part harmony, just like many country and Tejano acts.
I hear what you're saying, about having a hard time distinguishing harmony from melody. Sometimes it's like the word "irony". I can't explain it, but I know it when I hear it.
For examples of harmony:
1. "If I Fell" - Beatles. The lower line is the melody, and the higher line is the harmony.
2. "Drive My Car" - Beatles. Now the higher line is the melody, lower line the harmony.
If someone were to hear these records for the first time, and you asked them sing them back to you, the vast majority of the time people will automatically sing the melody. I have
no idea why that is, or how it works. However, I have seen this happen time and time again, whether it's in a cover band or church choir. People gravitate toward the melody. Some people will sing the harmony, if you sing it loudly enough, and close enough to them, so they can pick it out. Stop singing the harmony, and they'll revert back to the melody.