"Check-in" is the legal definition which the airline industry -- and many governmental bodies -- give to the process of committing to the ticketed flight. Therefore, the purpose it serves, in the different ways airlines allow check in, is to set the standard for the legal duties and responsibilities to carry ticketed passengers (and checked baggage) on that flight.
There may be added steps airlines can require (i.e., presentment at the boarding gate by a specified time) in order to set a standard of eligibility for other benefits (i.e., denied boarding compensation).