"Would "We didn't start the fire" by Billy Joel qualify as Folk?
"Down by the river" by Albert Hammond" ?
"The grave" by Don Maclean?"

No, no and no.

This is because these songs have not been through the PROCESS involving (i) continuity which links the present with the past; (ii) variation which springs from the creative impulse of the individual or the group; and (iii) selection by the community, which determines the form or forms in which [they have] survive[d].

This does not mean (deep sigh!) that these are 'good' or 'bad' songs - it just means that they are not folk songs, that's all, and, hence, not relevant to this discussion.