Strictly speaking, Bartelby does not give "web site" as an alternatibe, but "Web site" (with a capital "W").
See also Chicago Manual of Style
<<A lot of people are writing "website." A lot of people have come to prefer "website." But formal usage still calls for "Web site," [. . .]
We at Chicago are very particular too, and we recommend "Web site."
[. . .] generally, I would recommend "Web site" for formal writing, but "website" for informal writing or friendly writing. Unless, of course, you prefer "Web site" even when youre being friendly.>>
Unfortunately, LEO's usage classification does not offer "formal" and "informal".
I, personally, adhere to CMS usage in this case -- but, then, I also write "Internet".
The Economist style guide has "website", but I don't think it is an AE/BE thing.