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, in addition to not being an official ruling (which is good, considering Mearls' tweets rarely have any basis in the rules).
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 description of the paladin's Aura of Protection feature 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.
And the description of the Aura of Courage feature 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.
Rules designer Jeremy Crawford addressed a question about the area/range of a 10-foot paladin aura in this pair of unofficial tweets from March 2016:
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 Crawford's reasoning in this series of unofficial tweets from May 2018 as to why the centaur and minotaur playable races (originally released for playtesting in Unearthed Arcana: Centaurs and Minotaurs at the time) 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 5×5 area on a grid (with 5-foot squares); for a Large creature, that same aura is a circle over a 6×6 area on the grid.
This is a DM call.
As observed in another answer from Thomas Markov, there are two ways of reading this.
The first reading is literal: the Fireball is centered on yourself, hence the point of origin is the center of the square in which the characters is. The situation is depicted in the below figure, where the green circle is the Sorcerer, the red circles represent other creatures and the small circle the point of origin of Fireball.
The second reading applies the rules for using a tactical map from the DMG and the rules from XGTE:
Choose an intersection of squares as the point of origin of an area of effect, then follow the rules for that kind of area as normal (see the “Areas of Effect” section in chapter 10 of the Player’s Handbook). If an area of effect is circular and covers at least half a square, it affects that square.
In this case, there are 4 intersections around the square in which the Sorcerer is: then it has to be decided from which intersection the Fireball originates. The 4 situation are depicted in the figures below: pay attention that this choice affects in different ways the creatures around the Sorcerer.
North - east corner
South - east corner
South - west corner
North - west corner
It is up to the DM deciding who picks the point of origin: the player or themselves.
The scroll's effects can cover either four or nine Medium creatures, depending on how you interpret it.
Normally, effects that are described with a radius originate from a grid intersection. When the origin is defined by a creature, such as the caster of a spell, or as in this case, a creature reading a scroll, then the intersection is one of the intersections within or adjacent to that creature's space (the corner of their square for a Medium-sized creature). So if you take the "5-foot radius cylinder" as the key part of the spell, then the effect should cover four squares. One of those would be the square of the creature reading the scroll, and three of their friends could fit in the others. This is probably the best interpretation of the rule, though the description of the effect does it's best to make it more confusing by describing the effect as remaining "centered on you" as you move. Probably that just means you can pick a new corner for it each time you move.
The alternative interpretation is that the radius of the effect of a Scroll of Protection is supposed to be interpreted as a distance from the reader of the scroll, not a radius from a point in space. Since a Medium creature occupies a full 5-foot square in combat, there are 8 squares within five feet of them (forming a 3 by three grid with the creature's square in the middle). There are some spells, like the Thunderclap cantrip, that have areas of effect that work this way, and this is also how a Paladin's Aura of Protection works.
The spell Ice Knife as originally published (in the Princes of the Apocalypse adventure, and the free Elemental Evil Player's Companion that was released alongside it) used a radius to define the area of effect for its secondary explosion. But it was changed in an errata to use a distance from the targeted creature instead.
Unfortunately the exact intentions of the designers of the Scroll of Protection are not clear, so the DM of your game will need to make a ruling on how it works at your table. You could play by the strict RAW and use a radius that's centered on a corner of the reader's square. Or you could apply a similar ruling to the Ice Knife errata and change the five foot radius cylinder to a five foot range around the reader of the scroll, which might make the effect easier to understand.
I don't think either version is going to dramatically change the power level of the scroll. A larger area is not necessarily better, since you may want to be using the protection it offers to flee though a room full of the creatures you're protected from. Having a larger area means you'll have a harder time fitting through gaps in between the enemies (since you can't push the AOE against them without ending the effect). Similarly, with the larger area, the creature at the center (who read the scroll) can't easily reach the edge of its effect, and so they probably can't fight anyone outside unless they have ranged attacks or a reach weapon.