If I hold at a battlefield, move units out, then move units back into that battlefield that I held at, does it start a showdown?

Yes, moving your units back into that battlefield will start a non-combat showdown.

According to RiftJudge FAQ #7022, FAQ #7447, and Rule 323.6, here is the breakdown of what happens:

1. Losing Control: When you move your last unit out of the battlefield to your base, the battlefield becomes unoccupied and uncontrolled.
2. Contesting: When you move your units back in, you are moving into a battlefield you do not currently control. This makes the battlefield Contested (Rule 445).
3. Showdown: Because the battlefield is contested, a non-combat showdown will open during the cleanup phase following the move (Rule 344.2).

Important Nuances:

• No Scoring: You will not score another holding point or trigger conquering effects (like those from specific cards) at that battlefield for the rest of the turn, because you already scored it earlier (Rule 464, FAQ #7447).
• Non-Combat: Unless your opponent also moves units into that battlefield simultaneously, this is a non-combat showdown. Per Rule 348.2.a, since you are the only player with units present, you will regain control of the battlefield once the showdown closes.