Unless either Arlekino or Coaster is poorly worded Arlekino won't trigger off of Coaster like Multifaker does off of Spoofing.
You use the search effect of Spoofing when activating the card, which is why Multifaker can trigger, since a Trap Card was activated.
If you use the effect of a Spoofing that's already face-up you cannot summon Multifaker. For that same reason you cannot summon Arlekino off of Coaster alone.
When activating Coaster you target a monster to which it equips on resolution. Then, on a new chain you can activate Coaster's effect to add Arlekino. At that time no Trap Card has been activated, only the effect of a trap which has already been activated in a previous chain. You then need to activate another card to summon Arlekino.