What makes a jet private? Should Trump's Boegin 757 count as one? What if an airline is flying a jet with no passengers? Cargo jets?
The same thing that differentiate a private car from public transportation or freight, I would think. This distinction isn't a particularly novel problem.
We don't differentiate these in any significant way. Do buses in your country pay different rate for fuel?
There are vans carrying 6 people on international routes in Europe, is this public transport? Private? Anyone can book it.