That’s right, Webb isn’t a Dixiecrat or a conservative. He’s simply an old-school, Harry Truman or JFK-style liberal, as opposed to a leftist radical like most contemporary Democrats. In other words, Webb only recently figured out that if you’re a patriot and don’t apologize for being white (Appalachian Scots-Irish in his case), and if you’re not ready to grovel at the feet of blacks and other “minorities”, the Democratic Party doesn’t want anything to do with you.
Some of those Dixiecrats were hardly conservative, just old-school New Deal liberals who happened to be against civil rights.
As for Harry Truman and JFK-style liberals, many of their goals have already been achieved, so it's hard to say what their old-time liberalism would mean today.
When people say that JFK would be a conservative or a Republican if he were around today (which I don't really think, if only because it would mean bucking the rest of the family), what would that make Webb?