Is playing a unit from your hand to base considered a move?

No — playing a unit from your hand to your base is not a Move.

A Move is defined as a permanent changing position between two Locations on the Board (Rule 420.1, 446.1). Your hand isn't on the Board, so going hand → base is a zone change, which "does not in itself constitute a Move" (Rule 446.2). That means playing a unit from hand won't trigger "when I move" effects, apply Contested, or count as a Standard Move — even though the unit ends up on the board (FAQ #5660).