If we use the definition you seem to be to allow Woody to fit in a strict definition of folk music, you make ethnic music as folk/traditional music invisible.
Of course, there are music subcultures which aren't defined by ethnicity, or by cross-fertilization between ethnic groups (more common in North American music subcultures than many Old World (regardless of continent) music subcultures.
Techno/rave music is a subculture. Hip hop is a subculture. But there isn't a *tradition* of performance of those kinds of music from generation--that is the conventional definition of tradition, ie that it is multi-generational and handed on through the subculture in question.
Which is why I like Frank's definition so much. It isn't hard to eliminate many contemporary forms of music based on either the subculture argument or the tradition argument. If a genre of music lacks both, it isn't considered folk or traditional by people who know and are familiar with folk and trad music. If a genre of music lacks an ethnic basis, it doesn't mean it can't be considered a folk tradition though, as has been pointed out. But that (lack of ethnic roots) may well prove to be an exception in the long run (ie after several generations).
Fairly substantial numbers of people who share roots in the specific subculture have to care enough about the music to hand it on to the next generation. Historically, this has been done through musical families and neighbors. The contemporary world has changed that somewhat because of transportation and communication technological developments, but not near to the extent that some people imagine.
Its not all that easy to start a new music genre in a subculture and sustain it over multiple generations. Cajun music is a good example of this, as are a good number of other music genres.