What opens a chain outside and inside of showdowns?

Incorrect response in the first paragraph

One thing opens a chain everywhere: an item becoming Pending. Three sources — playing a card, activating an ability, or a triggered ability becoming Pending (Rules 328, 401.1). When it isn't your turn you would need a triggered ability or a showdown to be able to start a chain; only one chain exists at a time, so "starting a chain" only applies when no chain exists yet (Rule 330). Outside a showdown (Neutral Open): the default opener is the turn player in their Main Phase, playing a card or activating an ability. Note the state immediately becomes Closed once that happens, so follow-ups need Reaction. Triggered abilities can add items in any state, on anyone's turn. Inside a showdown: the player with Focus may start a chain with a card or activated ability that is legally timed — during Showdown Open that means Action or Reaction by default; nothing else is playable (Rule 347.1). Each spell played this way creates a chain as normal (Rule 342.1), and once a chain exists only Reactions can be added. Triggered abilities still land on the chain (e.g. attacker/defender triggers form the Combat Chain when combat opens). One subtlety: a Reaction can be the first item of a chain, since Reaction grants all Action permissions plus Closed-State play.