Does playing a spell start a new showdown?

No — playing a spell starts a Chain, not a Showdown (Rule 354.1). A Showdown is a separate structured window that gets Staged during a Cleanup whenever a battlefield becomes Contested, then begins from a Neutral Open State (Rules 344, 323.7, 323.11).

The common mix-up is with playing a unit: if playing a unit to a battlefield applies Contested, that battlefield will have a Showdown Staged in the next Cleanup — the Showdown comes from the Contested status, not from the act of playing the card.

The same applies in reverse: playing a spell during an ongoing Showdown doesn't start a new one, it just opens a Chain inside that Showdown and play continues until both players pass Focus in sequence (Rule 347.2.a).