http://definitions.uslegal.com/p/public-transportation/
Public transportstion refers to all service involved in the transportation of passengers for hire by means of street railway, elevated railway, subway, underground railroad, motor vehicles, or other means of conveyance generally associated with or developed for mass
surface or sub-surface transportation of the public, but does not include any service involved in transportation by taxicab, airport limousine, or industrial bus.
http://en.wikipedia.org/wiki/Public_Transportation
Public transport (also public transportation, public transit, or mass transit) comprises passenger transportation services which are available for use by the general public, as opposed to modes for private use such as automobiles or vehicles for hire.
Public transport services are usually funded by fares charged to each passenger, with varying levels of subsidy from local or national tax revenue; fully-subsidised, zero-fare services operate in some towns and cities.
Public transport can consist of subways, trolleys and light rail, commuter trains, buses, van pool services, paratransit services for senior citizens and people with disabilities, ferries, water taxis, or monorails.[1]
Nowhere in the above are airlines mentioned, in fact, the first definition is specific in non-air travel.
I think this debate is really more about perception than definition. Many people's perception when they hear the term public transportation is that is government funded, at least that is what I think. City buses jumps to mind. Whether this is correct or not, is obviously debatable.
I think you gentlemen should just realize you will not be changing each other's minds and move on to a new battle.