A couple of additions to the discussion......

"Songster" was a standard synonym for "singer" in the twenties, and while Mance Lipscomb indeed used the term Variety also used it for singers like Bing Crosby. It was only in the 1960s that blues historians reinvented it as a term for musicians like Lipscomb, John Hurt, and Leadbelly, who sang blues but also a wide variety of other material. However, it is important to remember that what really set those three apart was not the breadth of their repertoire, but rather the fact that they were "discovered" by people who were interested in documenting that breadth. Plenty of other musicians whose repertoires may well have been equally broad have been classed as pure blues singers simply because that was all that the people who recorded them wanted to record -- Robert and Tommy Johnson being prime examples. And this was not just because of the prejudices and fashions of the 1920s. One of the most prolific blues producers of the 1960s recently told me, in a discussion of Skip James's recording of Hoagy Carmichael's "Lazy Bones," "I hate that stuff. All those guys always wanted to record their version of 'Honeysuckle Rose,' and I'd have to tell them to stick to blues."

On Josh White, he was a popular gospel and blues performer on "race records" in the 1930s before ever being discovered by the white folk audience. His son, Josh Jr., does indeed remain active and is a fine musician. However, he is in no sense more of a blues and jazz singer than his father. Josh Sr. recorded with accompanists like Sidney Bechet and Mary Lou Williams, as well as a final session with Sonny Boy Williamson. Josh Jr. is more of a sixties-style folksinger a la Harry Belafonte, though he also does nice versions of his dad's stuff.
(And for more on Josh Sr., I wrote a biography of him, "Society Blues.")