Here in the NE its still quite common to see 'frankfurters", "franks", or "weenies/weiners" in addition to "hot dogs" written on menus. I call them whatever pops into my mind when ordering, LOL. The old Germantown aka Yorkville, still has a presence in Manhattan so when the mood hits we stop there to buy various sausages, smoked pork chops, and various sides and condiments. This is how Schaller + Weber still located in the heart of the old Germantown, sausages look like:
https://schallerweber.com/shop/sausages/
and I can say that the shape of knock/knackwurst differed in Germany/Austria and the German sections of Switzerland by length but not by meat content (always a mix of pork and veal), colour or diameter. Franks are made with all pork no matter the region or country.
I canna say what happens in other parts of the US when it comes to German sausage making and although I haven't eaten at Biergarten, I have grabbed a sausage from Sommerfest where only 2 types are offered: frankfurters (German style) and bratwurst. I agree with your German teacher, the American dog is a combo of seasoning from the 2 different sausages

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