Note.— The final approach path vertical path angle (VPA) is safeguarded against the effects of low temperature by the design of the procedure.
when the temperature is within promulgated limits, the flight crew should not make compensation to the altitude at the FAF and DA/H; and
Technically, these are saying the same thing....
and yes, when a temperature correction is required, the pilot is responsible, but it must be cleared with ATC or you dont use the procedure.
It is discussed in detail at the link Capt S provided.