Air carrier Captain reported a fume/smoke event after takeoff. After troubleshooting flight crew executed a safe air return.

Narrative

Arrived at ZZZ maintenance facility to reposition aircraft to ZZZ1. Accomplished full originating/acceptance checklists. Completed fully all normal checklists prior to takeoff. No unresolved discrepancies. Normal takeoff with flex power. (However; we did not shut down APU until just prior to take off; thus PACKS were on APU till that point). Immediately after 'Positive rate; gear up; speed mode' the cockpit filled suddenly with smoke of moderate density. The smoke had a distinct smell that comes only from the PACKS and definitely NOT electrical or fuel. We discussed and shut off PACKS and dumped cabin to clear smoke as we were at low altitude. While climbing we advised Tower of the smoke and requested remain in the pattern for return to field. He asked if we were declaring an emergency and we replied NO; NOT AT PRESENT. ( the same reply we gave to all subsequent questions ). We referenced EPC and QRH. On downwind we got a SMOKE FWD LAV warning and audio. First Officer got up to inspect lav and found it was only from the residual cockpit smoke and no fire was present. Also checked aft lav. Warning indication on EICAS went away. Returned to cockpit discussed all relevant circumstances and determined the threat was managed and ZZZ1 (10 minutes away) offered better maintenance and rescue and fire facilities. Made low approach to runway and coordinated with ATC and proceeded to ZZZ1 uneventfully. (PACKS remained OFF).