"There's a lot of crossover with the Twa Brothers, to the point where it's not obvious that 'Edward' (or whatever we're going to call it) really was a separate ballad."

We had a discussion about the relationship between 'Edward'(13) and 'The Twa Brothers'(49) recently - my contribution starts here.

I see several distinctive features in 49 that aren't in 13, and suggest that the presence of "What's that Blood?" ansd "When Will You Return?" verses in some texts of 49 (though remarkably few of the American oral versions) probably represents an interpolation. I'd be interest to know what Steve G thinks.

"no real connection with Lizzie Wan"

Again if you look at the oral versions - which generally lack all that stuff about heads and bodies getting chopped up, or indeed any description of the murder at all - both 13 and 51 usually consist of "What's the Blood?" and "What Will You Do?" sequences and are often virtually indistinguishable.