Interesting question and discussion, but I agree with Bob that the answer is both - songs can be generic and specific; sometimes even in the same song. The genres I write for are strong on storytelling ala Johnny Cash, Guy Clark, et.al, but the stuff that makes it on the radio is pretty generic. Has anyone else noticed that often times the songs you find yourself really liking and listening to on an album are not the singles getting airplay? Happens to me quite often.
I'm definitely still in the process of learning to write good ballads and story songs, but a couple things I think I've learned are:
1) The detail is in the imagery, the little things that sourround what the main subject of the song is. The song may be about a romantic encounter, but what keeps a listener's attention is what they might recognize from their own experience - the smell of a lover's hair, the of a touch of a hand on your face or the scent of perfume. Everybody understands the general thrashing around and physicality of the moment, but it's the detail of all the peripherals that keep people listening.
2) What you don't say in the song is as important or more important than what you do say. Once you've given people a detail like "caught a whiff of her perfume", you don't need to say Chanel #5 or whatever; let the listener add their own little detail. It's kind of like drawing a picture and letting the listener color it in.
Generic songs are great, especially if you want to hear your song on the radio. There's nothing wrong with providing people with background or mood music and it takes talent to write those kinds of songs. Lots of songwriters write for specific markets, but I think you'll also find that we write some things for the simple fact that they make good songs. Songwriters also change genres and write for different markets during their careers. That's how they survive 40 - 50 years.