At what point does an aircraft become an airliner?
I think there's a misunderstanding here that airliner ("an airplane operated by an airline") is directly derived from airline. While it is a reasonable definition, "airliner" predates airline. (Mirriam Webster: airliner 1908, airline 1910). 

An airliner is a portmanteau of air and liner, a liner being any passenger ship plying between ports along regular "lines" (as distinguished from transient ships using those ports). By the early 20th century a liner was generally understood to mean any large, passenger-carrying vessel operating a route between major ports (think Titanic etc.). You'll notice how airliners travel between airports - nautical words and phrases were adopted in the early days of flight.

So an airliner should be


passenger carrying
large 
travels between airports, rather than having one fixed point of operation


It's a definition that still fits, and covers any commercial aircraft over a certain size. While there's not a specific point at which an aircraft becomes large enough to be called an airliner, I think it does impart some sense of largeness or capacity that an 'ordinary' plane does not possess.

(You might even say that a better definition of airline is "an air transportation system that operates airliners"!)