At which point can a power supply can rate itself at a certain current
A well-designed 9 Volt 5 Amp regulated power supply should deliver 9.0 Volts, and the output voltage should only vary a few millivolts with load currents from 0 to 5 Amps.

The 5 Amp rating means that the supply can deliver up to that current without damage or over-heating.

Cheap unregulated wall-wart supplies should deliver up to the rated current while remaining "close" to the specified voltage and often deliver significantly more than the rated voltage at low loads.  Only the maker knows what "close" means.