fATPL is a nickname, and it applies to Europe and perhaps some others. I don't know how Canada works, but in the US the ATPL exams cannot be sat until the hours requirement is met, so there is no such thing. The frozen ATPL is a CPL/IR with passes in the ATPL exams.
The CPL/ATPL difference is nothing to do with who you work for; it relates to what aircraft you can command. A CPL allows you to earn money from flying, and allows you to fly as crew of any aircraft (given the appropriate class or type rating), including commanding a single-pilot aircraft (although there might be further conditions, such as requirement for 700 hours for single-pilot IFR operations in the UK). A full ATPL allows you to command a multi-crew aircraft.
In Europe most people skip the CPL exams, and do the ATPLs. In the US that is impossible unless you can achieve 1500 hors on a PPL.