The frankfurter is indeed a sausage and must be put on a certain style bun to become a legit hotdog. I believe simple usage determined that the sausage may be sold as a "hotdog."
A frankfurter is not a hotdog until surrounded by a poppy seed bun, dressed in bright green relish, mustard, mayo, with other condiments such as sport peppers, cucumber, possibly sauerkraut, and never, ever touched by catsup.* Ketchup? (Children may add catsup/ketchup but, children have not developed the discerning taste which is required for the proper enjoyment of a well built hotdog.) It is at it's best when a Vienna Sausage wiener is used. The last is only my opinion of course. Nathan's isn't bad as are most other "Kosher" all-beef franks/wieners with skins/casings which snap when pierced by the tooth. Turkey, tofu or chicken do not a hot dog fit wiener make! Pork makes... pork sausage, good on a bun to be sure. But, not a hotdawg by the stretch of anyone's imagination.
And yes! The venerable "hotdog" is acceptable fare to many British citizens.
There is no room for debate here! Let the debate begin!
* Properly, one dresses their frank not the bun. Do not dress the bun. Dressing the bun is considered to be boorish behavior, or possibly, even a serious faux pas in many locales. Dressing the bun can and has led to social ostracism in certain circles.