It's a difficult area and subject to much disagreement. One useful (though by no means exclusive) criterion is the context in which the singer's involvement with the music has developed; where they learned their attitude toward it, and their basic style. Repertoire can come (and probably always has) from anywhere, but it may be that one distinction between the traditional and revival singer is whether or not they have grown up (physically or as a singer) in an existing and identifiable tradition. That's largely what forms attitudes.

To an extent, everybody carries tradition of one sort or another, but that isn't really the point. Class and profession aren't the issue (though many people have supposed that they were), but it's generally true that traditional songs have persisted best in coherent communities where they were a normal part of life. As to whether or not Mr Greaves should be considered a traditional singer, I'd say that it might depend on his background; on where he's coming from in his relationship with the material rather than where he got it.

Context again; it is, for example, quite beside the point that the Coppers sing from written texts. Equally, singing traditional songs doesn't of itself make a person a traditional singer, any more than a traditional singer makes a song traditional by the mere act of singing it (though that may be the start of a process through which it will become so). You will probably want to look at the way in which many traditional singers would effectively serve an apprenticeship in their art; Ginette Dunn, Fellowship of Song, is particularly useful on that (perhaps you've already read the book) and it also illustrates that many traditional singers have quite strong opinions both on their material and on its meaning to them and to others.

The terms are neither silly nor irrelevant; but they are limiting. There is considerable scope for further serious study of the mechanisms of transmission within the Revival, the extent to which it overlaps with identifiable and continuous tradition, and the extent to which it may be considered that new strands of tradition are emerging. This, after all, is a continuing process; tradition is made by time and continuity (as well as by change) and we can't just appoint ourselves to it. One does not "harm" the folk-process (a term almost as misused as the word "Celtic") by examining it, any more than we harm nature by studying the migration pattern of birds.