When a die comes up Terror in a fail, it says all Terror effects on the adventure and the mythos card in play activate. The mythos card rules say that they have an 'immediate effect' and a 'lingering effect'. Are both of these effects activated in such cases? We played just the immediate.
There is (currently) one and only one Mythos card that has a Terror effect. This is the only time the situation the rule on page 7 applies.
...in a Disturbed Mind
All Adventure cards gain the following Terror effect today, replacing any Terror effect they normally have.
Terror: Discard all terror dice.
To support this take a look at page 7 of the rules as it gives an example of a Terror effect. Apart from the one Mythos card mentioned above, no other Mythos card has the wording "Terror: ..." as shown on the example.
Also, under Resolving Mythos Cards on page 9, nowhere does it mentioned that either the Immediate Effect or the Lingering Effect applies to when a Terror effect is triggered. The rules on page 9 are clear as far as I'm concerned.
The immediate effect, which is on the upper half of the card, is an effect that takes place as soon as the card is drawn.
The lingering effect, which is on the lower half of the card,
provides special rules which either apply to the game until the
next Mythos card is drawn or take effect the next time a player
advances the clock to midnight and other At Midnight effects
So, I think it is clear that Terror effects do not trigger any of the other Mythos card immediate or lingering effects.