Timing: Flies and Spiders AND The Ring Will Have But One Master

[Konrad Klar]

Playable on a character facing a Spider attack. If the strike is successful, target character is not harmed and is taken prisoner at a Ruins & Lairs [ ]. During his untap phase, make a body check for that character modified by +1. If not eliminated, his player then makes a roll adding his body. If the result is greater than 15, the character is automatically rescued into his own company located at the rescue site.

Rescue-attack: Spiders — 3 strikes with 9 prowess.

Playable on the bearer of The One Ring. During each of his organization phases, one character (other than the bearer) in bearer’s company is discarded (of bearer’s player’s choice) along with all non-follower cards played with him. During his organization phase, the target character may tap to attempt to remove this card. Make a roll—if the result is greater than 8, discard this card. Cannot be duplicated on a given character.

For comparison:

Playable on a character facing an Undead strike. If the strike is successful, target character is not harmed and is taken prisoner at a Ruins & Lairs [ ] or Shadow-hold [ ]. Character must discard any rings along with his other items. At the start of each of his untap phases, make a body check for that character.

Rescue-attack: Undead — 3 strikes with 8 prowess.

Flies and Spiders AND The Ring Will Have But One Master do not specify an exact moment of triggering of actions they create.
Die roll actions may be targeted by other actions declared in response in the same chain of effects. But it is unclear (for me) whether the actions created by the two cards are declared at all.
Maybe the two cards are candidates for errata.

[Bandobras Took]

Since both specify they happen during a phase, I would treat them as actions declared by a passive condition that must be declared after resolving any actions that must be taken at the beginning of the phase and before declaring any actions that must be taken at the end of that phase.

I would also say that the resource player may choose exactly when within the aforementioned limits they wish for the action to be declared.

[Konrad Klar]

Are there any other examples of actions caused by passive conditions where player may choose an exact moment in which they will be declared?

Annotation 9: If a card specifies that an action is to occur as a result of some specific
passive condition, this action becomes automatically the first action declared in the
chain of effects to immediately follow the chain of effects producing the passive
condition. The passive condition must exist when this resulting action is resolved in
its own chain of effects, or the action is canceled. Note that actions in the strike
sequence follow a different set of rules.

Annotation 10: If more than one action is required to be the first action declared in a
chain of effects, the player whose turn it is chooses the order in which they are
declared. No other actions may be declared in this follow-up chain until the multiple
required actions have been declared.

I do not mean a choosing of order in chain of effects in which the actions are declared, described in Annotation 10, but a an exact moment it which the chain is started.

Annotation 9a: If a card is required to be discarded by some passive condition, the
card is discarded immediately when the condition resolves, not in the following chain
of effects.

Or moment when the action is executed (without chain of effects) in case of The Ring Will Have But One Master.

[Bandobras Took]

A New Ringlord.

Evil Things Lingering.

The Balrog (Ally).

RW mode cards.

Padding Feet.

All of these have actions that must be taken when the conditions are met, but the use of "during" allows for the choice of exactly when during the specified window the action will be taken.

[Konrad Klar]

I would say except RW mode cards an Padding Feet - player taps/rotate the card once site is entered (and condition is fulfilled).
And if RW mode cards would not be discarded immediately when conditions are met then they could be utilized to make attempt to move to another non-Darkhaven.