CRJ-200 First Officer complied with non-standard ATC departure instructions to turn left direct to a fix which resulted in them receiving a Ground Proximity Alert.

Narrative

On the last leg after a four day trip; in VMC conditions; the Initial IFR Clearance stated 'Fly runway heading'. Pilot-Flying (FO) briefed terrain; obstacles avoidance by using ODP. Takeoff clearance given by ATC included 'Left turn direct ZZZZZ; traffic is in 2 miles final'. As a result; PF (Pilot Flying) executed a left turn at 500 ft. under VMC; verifying full obstacle clearance. During the initial climb the PF's radio altimeter was unreadable (reading changed rapidly) and uncorrelated with the Pilot-Monitoring (Captain)'s radio altimeter which showed solid climbing readings; and a single EGPWS alarm was noted (Whoop-Whoop; Pull Up); correspondingly both pilots verified again clearing of obstacles and terrain proximity. Later; during the approach to the destination; radio altimeter showed reading only on the PM (Pilot Monitoring) (Captain) side; which were uncorrelated with the current actual altitude. Traffic clearance concern: If takeoff clearance wouldn't include the word 'direct' but 'on course'; the correlation between the 2 miles on-final traffic to the left turn on course wouldn't be to expedite the turn by the Pilot Flying for traffic avoidance (also; in case of a Go-around by the landing plane). ATC clearance should be stated: 'Left turn on course ZZZZZ' / 'Left turn on course' would be sufficient for the Pilot Flying to execute his climb planning as planned during the preflight.