Quite right, Texas Alexander sings a different song, unless there's another recording of the correct title and lyrics. It's unlikely that there is, however, because 1) the song seems to have originated from Kentucky and 2) the original lyrics were about a woman and I doubt that a Texas bluesman would have changed the gender for an adaptation.
See Alan Lomax's comments about the 1937 recording at, among other places,
http://www.fact-index.com/t/th/the_house_of_the_rising_sun.html
although I've seen other claims of earlier country recordings under slightly different titles but haven't researched these. If anyone does, please let me know what you find!
Recording is in print
http://www.amazon.com/exec/obidos/tg/detail/-/B0000AUHRE
The place of origin clearly makes it country, but fellow countryman Doc Boggs played some pretty sad ballads that might be called blues. Rather than getting hung up on catagories, suffice it to say that it originated from white folk, not black.
How New Orleans ties into Kentucky is another mystery for the musicologysts and historians.