What is the advantage of defining IP3?

what is the advantage of defining IP3, why we don't just measure the fundamental & 3rd order component of output and divide them by A and A3 (A:input signal amplitude) respectively.
I mean :apply two input signals of amplitude A and measure the output IM
then since the IM3 is proportional to A3
and fundamental to A we can divide our measurement by A3 and A
obtain two quantities representing the PAs linearity

Why IP3?

As you said, IM3 is related to the input signal power level
on the other hand, IP3 will not change with different input signal power

therefore, it is more suitable for comparison

Well if we divide IM3 by A3 we will have 3/4*α3
(if you remember the formula! )
and it doesn't changes with input power as well
so we really don't need IP3
we have "3/4*α3" and "α1" for every typical amp. and these can be compared with other amps.

so when you give a IM3 value, you also need to specify the input power level.And still some calculations

why not just IIP3?Doesn't it make life easier?

Actually when you give IM3 you have to specify the input level.

If the amplifier is in compression the IP3 calculation is not valid anymore.