To play Devil's Advocate:
They do not have step 5: apply damage, step 6: apply other effects.
Applying damage and other effects is one single step. They couldn't really say "apply other effects and damage" now could they? So, both things occur at the 'same time'.
It's one step, but you read a power in order. And in this case, as this isn't an "Effect:" in the defined game term sense of reading a power which is listed in a sequence It could reasonably be stated that the components of the trigger 1) Hitting (before damage and most other effects) and possibly 2) Invigorating Keyword (before pretty much everything) are met before damage, conditions and most "Effects" are applied.
Now, as has been stated by Cadfan, if you modified the damage whenever you'd hit, you would always deal the extra damage (except for a non-invigorating daily that damages on a miss). So it seems unlikely they'd make something like "when you have thp do extra damage" if NOT dealing the extra damage would be the exception.
It seems just as unlikely they'd include a bonus that you wouldn't get enough to be worth giving up better armor. And seems unlikely, either way, is about as meaningful in a discussion as what the rule actually is as claiming to know the intent of something, when WotC employees who designed material can't always explain the intent of changes to their original design (Double Weapons) or their clear explanations of intent are contradicted officially (Rain of Blows).
The credibilty of statements (from non-WotC staff) that claim to know the intent or that some intent is implied and that intent they "know" or "is implied" changes what the rule actually says and means are as credible as statements that the intent - "known" or "implied" - meant Dwarven Weapon Proficiency didn't grant proficiency in superior hammers and axes or Swordmages couldn't reap the superior benefits of Warding and Versatile.
They didn't re-write the paragraph regarding the bonus damage, and the actual errata is:
"Whenever you hit an enemy with a melee or a close attack, you gain temporary hit points equal to your Constitution modifier, plus any temporary hit points normally granted by the power.
If you use an invigorating fighter attack power and miss every target with it, you gain temporary hit points equal to your Constitution modifier."
In the previous version, they specifically felt the need to include language that allocated the temp hit points after the attack against the battlerager was resolved. No such language is present here. That may or may not mean something. It may mean they meant for the HPs to be granted immediately after the hit. It may mean there is something in the language of the ability that means something to the author or editor that makes it unnecessary or redundant. It may have been removed due to formatting restraints on reprints.
They might just clarify it as "if you had thp before you attacked" ... since the damage boost is now something that you get BECAUSE you still have thp (and thus when you hit, you are getting fewer thp than you normally would.)
They might, but they didn't in the update. And there is no way of really knowing what the answer they give, if any, is going to be. It may specifically follow the RAW, it may create a new game term, it may claim that no errata or FAQ is needed because things happen in a certain way even though that way isn't clearly defined in the books.
Certain phrases and words seem to have a specific meaning to people who work for WotC that isn't explicitly stated in the books.