Paladin auras "originate" from their entire space, so the radius extends from the edge of their space
In this case, Mike Mearls' interpretation in your second link is incorrect (or at least misleading).
All of the paladin auras don't specify a "point" of origin; instead, they merely specify that certain effects apply to creatures "within 10 feet of you". (At 18th level, the range of your auras increases to 30 feet.) In essence, it's an "area of origin" rather than a single point.
For instance, the Aura of Protection states:
Starting at 6th level, whenever you or a friendly creature within 10 feet of you must make a saving throw, the creature gains a bonus to the saving throw equal to your Charisma modifier (with a minimum bonus of +1). You must be conscious to grant this bonus.
The Aura of Courage says:
Starting at 10th level, you and friendly creatures within 10 feet of you can’t be frightened while you are conscious.
(Most of the subclass options also grant their own auras at 7th level.)
Logically speaking, it would make no sense to say that a creature right next to a Large or Huge creature is not "within 10 feet of it". Thus, an aura is an effect that does not have a point of origin; it originates from you and your space as a whole.
Jeremy Crawford, official rules designer for 5e, addresses a question about the area/range of a 10-foot paladin aura here on Twitter:
Which is correct representing the area of paladin aura (10 ft)?
Is each square 5 ft., and which aura are you talking about?
Yes, each square is 5 ft. I meant Aura of Protection and Aura of Courage. Thanks.
Aura of Protection/Courage extends 10 ft. all around. (None are correct, unless your circles are polygons.)
In that example image:
- option A treats the aura as if it originates from the center of the
- option B treats it as if it were a spell, originating from a corner
of the grid adjacent to the creature
- option C treats it as if it extends from the edge of the creature's
space, but inaccurately depicts the circular aura
Crawford's response suggests that C is the closest to being correct, but is wrong because of the way it depicts the area of the aura as a polygon instead of a circle.
(Also, even if the DM did decide to have the aura be square instead of circular since they're playing on a grid, the diagonals in C are too short.)
This interpretation is further supported by Jeremy Crawford's reasoning here as to why the centaur and minotaur playable races (from Unearthed Arcana: Centaurs and Minotaurs) should mechanically remain Medium like all other playable races instead of Large:
Visualizing one of @JeremyECrawford's reasons why Centaur and Minotaur should not be Large PCs: a 5' aura is 50% bigger and a 10' aura is 33% bigger.
That's exactly right.
Is it safe to assume we shouldn’t expect Large PC races?
In short: Paladin auras extend from the edge of the paladin's space, not from any individual point.
For a Medium creature, a 10-foot aura is a circle over a 5x5 area on a grid (with 5-foot squares); for a Large creature, that same aura is a circle over a 6x6 area on the grid.