Diffusion current in semiconductor while applying an electric field

When you don't apply an E field the drift and diffusion current balance ( both flowing in opposite to each other). The diffusion current is quite small and does not depend on applied E. When E is applied the drift current changes and therefore the net flow of current across the diode is no longer zero. If the diode is reversed biased the diffusion current dominates. When it is forward biased the built in potential is lowered and the drift current is larger than the diffusion current.

You are right that the absence of a diffusion current implies no statistical trend in the carrier density from point to point. This however does not exclude the possibility of a non zero expectation for the carrier current<i> density, which would arise from an applied field.