When do targets need to be specified when casting spells - on cast or on resolution?

Ruling: Targets must be specified when casting a spell or activating an ability. Anything you choose (that isn't automatically determined by characteristics) is a target and must be chosen on cast.

Sequence:

• Split the effect into immediate effect and follow-up effects by looking for "do this:" in the text
• Identify all things in the immediate effect where you have a choice (even theoretical choice counts)
• Those are targets and must be chosen when playing the spell/activating the ability
• When it resolves, check the "do this:" part - anything after creates reflexive triggers
• Reflexive trigger targets are chosen when the trigger goes on the Chain (at resolution)

Nuances:

• Cards like Falling Star and Icathian Rain don't target anything; they only create reflexive triggers
• "Choose" is not required wording for something to target
• Some cards with "choose" don't actually target (like Cull)
• What matters is whether you have a choice in theory, not in practice (e.g., "Kill a unit" targets even if only one unit exists)
• "Kill the unit with the least might" doesn't target because you have no choice