The difference between 'transfer' and 'transit' in the context of airports and train stations
Solution 1:
According to several sources found by googling "transit vs transfer";
Transit means people on the same flight/plane/airline. For example when a plane lands to refuel.
Transfer means people switching flight/plane airline.
A transit would be cheaper because you, the passenger, don't really need to do anything at the airport like checking out and in.
Extrapolating logically, I would explain them as halfway switch and endpoint switch, respectively. Transit visa and transit flight are two entirely different things. A flight means the flight is still 'in progress', but momentarilly paused. This means you're not actually "in the country", but on the flight. If you transfer between flights, one flight ended and another one starts... but your JOURNEY has not ended, it is just paused, i.e. in transit. But because you're not on either flight at the moment, you're in the country, and thus you need a visa.
Not needing a visa also ties in to absence of a (flight) transit fee, I suppose.
Solution 2:
In the context of travel ...
A "transfer" is essentially moving from one item to another item. So it would apply equally to going from a hotel to an airport, as well as going from one airplane to another airplane.
The term "transit" applies more to staying temporarily before continuing on to your next destination. You would "transit" an airport if you fly from A, stop for a while, then continue another flight to B. You "transit" a subway station when you stop to change from train C to train D.