I think that the reference of private use in F/O's description should be interpetated as for personal use.
We fly "corperate" (private aircraft owned by company for both company and personal use of owner) under EASA rules and our local CAA applied for us to ICAO in Montreal and after a while we were issued our requested callsign and ICAO 3 letter code for flightplan filing.
But as usual I guess that every local CAA has his own interpretation of the rules and since ICAO only accepts requests from local CAAs (found out that one the hard way), you are bound by what your local CAA desides.so in practice this means 27 different sets of rules.