As the Rules Tips Blog editor, I can confirm that the article is to cover the common case of trying to use an activated ability to process a card in exile in the middle of Displacer's ability resolving. It is not intended to cover every possible corner case (which this definitely is), and isn't meant to be a definitive statement that something can never happen.
Players will often ask, "Can I {eldrazi processor activated ability} that card before it comes back?" and that's what this blog post is meant to address.
As already noted, yes, the replacement effect of Ulamog's Despoiler could process the Banisher Priest as it enters immediately after Banisher Priest is exiled, before it would be returned by the Displacer's ability.
I don't see a creature dying between step 4 and 5, so now sure how you reached step 5 at all. Before damage, Sengir Vampire has one damage marked on it from Ashmouth's ability, then you add 1 more damage from Vithian Stinger. That leaves a 4/4 with 2 damage marked on it, and no creatures have died.
One Ashmouth Hound and Sengir Vampire deal damage to each other, both die at the same moment from lethal damage (this triggers Sengir Vampire's ability because it died at the same moment as the Hound, though that trigger is irrelevant since it can't add a counter to a creature no longer on the battlefield). This triggers Butcher, and you sacrifice Stinger. Stinger's death doesn't trigger Sengir, because it's already been dead for a while now.
Yes it's possible to Chord for 5, using the Birds for either mana or Convoke yields the same result. You announce Chord with X=5, and total cost becomes 5GGG. You can pay that cost using your lands for GG1, and tap 5 creatures to pay for G4, or use lands and Birds for GGG2, and tap 3 creatures to pay for 3. The only thing you CAN'T do is use Birds as both mana sources AND tap them for convoke, since mana abilities must be activated before paying costs, and if the Birds are tapped when you go to pay costs, they can't be tapped for Convoke.
You're venturing outside the scope of the rules forum here. But it's both for flavor and for easy reference. "I like the abilities that happen when a land enters the battlefield..." or "I like those abilities that work when you have 7 more cards in your graveyard" or "I like those abilities that trigger when you target your own creature with a spell..." are all clunky ways to refer to Landfall, Threshold and Heroic abilities, respectively. This is not a new thing that started in Zendikar, and it doesn't really cause many problems, while it makes things a lot easier to talk about.
EDIT: Does this also mean that Battalion and Exalted technically aren't keyword either?
What exactly determines whether an ability word is or becomes a keyword?
Keywords are in rules text, ability words are in italics (just like reminder text, with the same amount of rules meaning). Battalion is an ability word (italicized, not a keyword) that links together "when this and 2 creatures attack..." abilities, while Exalted is an actual keyword ability with rules text associated with it.
A general guide is that ability words don't always do the same thing; they're just thematically linked by a common trigger condition (Heroic, Battalion) or "turned on" by the same game state (Threshold) or previous occurence in the game (Morbid), but each has its own unique effect. Keyword abilities always do the exact same thing (occasionally with only a numeric difference, such as with "Tribute X").
Yes, a creature token's name is the same as its creature type by default (i.e., unless the effect creating it gives it a different name, like Dark Depths does). Both tokens are named "Wurm" so both will be destroyed by Maelstrom Pulse.
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Players will often ask, "Can I {eldrazi processor activated ability} that card before it comes back?" and that's what this blog post is meant to address.
As already noted, yes, the replacement effect of Ulamog's Despoiler could process the Banisher Priest as it enters immediately after Banisher Priest is exiled, before it would be returned by the Displacer's ability.
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If you mean it would cause you to gain life and therefore would be modified by Rain of Gore into life loss instead, that's correct.
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One Ashmouth Hound and Sengir Vampire deal damage to each other, both die at the same moment from lethal damage (this triggers Sengir Vampire's ability because it died at the same moment as the Hound, though that trigger is irrelevant since it can't add a counter to a creature no longer on the battlefield). This triggers Butcher, and you sacrifice Stinger. Stinger's death doesn't trigger Sengir, because it's already been dead for a while now.
Hope this makes sense!
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Keywords are in rules text, ability words are in italics (just like reminder text, with the same amount of rules meaning). Battalion is an ability word (italicized, not a keyword) that links together "when this and 2 creatures attack..." abilities, while Exalted is an actual keyword ability with rules text associated with it.
A general guide is that ability words don't always do the same thing; they're just thematically linked by a common trigger condition (Heroic, Battalion) or "turned on" by the same game state (Threshold) or previous occurence in the game (Morbid), but each has its own unique effect. Keyword abilities always do the exact same thing (occasionally with only a numeric difference, such as with "Tribute X").
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TOP
Pharika's Cure
Heroic trigger that wants to put a +1/+1 counter on Hero
Alpha Authority
BOTTOM
So working from there, this is the outcome:
Pharika's Cure resolves first, then Hero is a 2/2 with 2 damage and it dies.
Heroic trigger resolves doing nothing.
Alpha Authority is countered.
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