In the early days, all seats were first class. (Source)
Postwar, some airlines started introducing an additional coach class--sometimes also called tourist class or economy class (the last name is today most common). See also "The Origin of Economy Class" (KLM Blog, 2019).
In the late 1970s (and after 1969 when the Boeing 747 was introduced), airlines saw an increasing need to also have a third intermediate class for business or emergency travelers.
Relevant information from The Financial Post (1978):
KLM was one of the pioneers with its FFF class (meaning full fare-paying facility) and most other airlines, sooner or later, followed with variations ...
British Airways ... is launching Club Class ...
"It's our way of recognizing the considerably higher fare paid as well as the quality of service demanded by the business traveler" ...
Pan Am quickly followed with ... Clipper Class ...
CP Air was working out details for something similar it expected to label "Executive-Economy Service." ...
Air Canada has no specific plans at this time, but ... acknowledges three motivations for travel and says "We have to meet those needs with three class of service." His three groups are first-class, business or emergency travelers who need flexibility and, "as we have seen this summer, those responding to price."
Airline observers agree it is only right and proper the business/emergency segment should get some sort of accommodation. It is much too large and revenue-productive to be ignored -- and growing ...
On the North Atlantic air route, the world's heaviest, an Iata survey shows that business traffic increased dramatically last year to 28.8% vs 21% in 1976 and 19.9% in 1975.
All of the above mentioned airlines were explicitly targeting this intermediate class of business or emergency travelers. However, none explicitly called it "Business Class" (but instead used names like FFF or Club or Clipper Class or Executive-Economy Service).
From my brief googling, it seems Qantas might have been the first to explicitly use the name "Business Class". From a 1980 Business Week ad:
The New Qantas Business Class. You get first-class service for about $900 less than the First Class fare to Australia. Our new Business Class is not just the old Economy Class with a few frills tacked on.
The term business hotel seems to be rare or non-existent in a search of Google Books for 2000 and before. So I'm guessing it was simply copied from business-class flights.
I had never heard of business taxis until reading your above question. Googling "business taxi", I find only about 200-300 results--many are irrelevant and those that are relevant seem to be for websites in Russia, Germany, Switzerland, Sweden, the Netherlands. So perhaps this term is mainly a continental European thing? (And again, I would guess that it was also simply copied from business-class flights.)