I hope that Rocky can take a breath for a second here and go along for a bit.
Names for knots are like common plant names. Since there isn't a clearing house or registration, names for knots get a little casual.
What arbos call a double fisherman's knot/hitch might go by other names in other rope disciplines. After looking at many knot books and many knot/rope disciplines, it seems [to me] that the most common use for the double fisherman's knot/hitch refers to the part of the rope configuration that makes "crossing round turns with a tuck". When it's used as a termination for a friction hitch it could be more properly be titled a sliding double fisherman's knot/hitch.
Budworth has been raked over the coals for repeating wrong names in many of his books. He's also been accused of repeating wrong information from other books in his books. Just because it's written doesn't make it universally "right".
Since we argue about seal/heal, head/top, false crotch/Friction Saver, it makes perfect sense that we butt heads about knot nomenclature too.
If you want to hear knot geeks Rocky, you should follow the International Guild of Knot Tyers. That organization is as close to a ruling body for knot nomenclature. If all arbos chose to abide by their decision for naming this rope configuration, then we wouldn't have to put up with the arbo geeks of the world.
In the end, it gets down to tomato or tomato, potato or potatoe [for the Quayle folks].
Tom