The DMG rules are consistent if you count the exact number of spaces, while measuring either way around.
In other words, if you have two creatures adjacent to a middle creature, then count both clockwise and counterclockwise. If the number of spaces (in either direction) equals the number you're trying to count, then there are that many spaces in between.
The trouble here is a geometric property that on a hex grid, Large and Gargantuan creatures are surrounded by an odd number of hexes. If you pick two of these hexes (as possible flanking locations), and count the spaces in between, then you are guaranteed to get different measurements clockwise versus counterclockwise.
However, it is possible to pick two surrounding hexes such that, if you count the in-between spaces both clockwise and counterclockwise, the measurements differ by 1. This is the closest way to split the difference (an odd number), so the DMG uses these measurements to determine flanking. Thus if a creature is surrounded by an odd number of hexes (i.e., they are Large or Gargantuan), then an adjacent attacker can flank with up to two other adjacent attackers.
When two Medium creatures are adjacent to a Gargantuan creature, there are 13 free adjacent spaces. If you can count 6 one way, then you must count 7 the other way. In this case, you counted the short way, but you still counted exactly 6, so they are flanking.
When two Medium creatures are adjacent to a Large creature, there are 7 free adjacent spaces. If you can count 4 one way, then you must count 3 the other way. In this case, you counted the long way, but you still counted exactly 4, so they are flanking.
By comparison, with Medium and Huge creatures, they are surrounded by an even number of hexes. It is possible to pick two of these hexes, such that you can count the same number of spaces in between both clockwise and counterclockwise; these are visibly the flanking positions.
As a side note, the second DMG quote only applies to a square grid. Otherwise you are dealing with creatures that don't have "opposite sides or corners".
So for the purpose of determining whether two attackers flank a Large (or Gargantuan) creature, you will have to count either clockwise or counterclockwise.
In the above examples, there are exactly 4 spaces counted in between the attacking creatures.
This is absolutely counter-intuitive for me, when you mechanically can't effectively hide behind a 5 foot tree against a non-moving attacker.
What you need to understand is that the mechanics of combat are intended to function as abstractions for what actually happens in combat. Diegetically, the combatants aren't literally standing in place, waiting for all the other combatants to finish their "turn", every six seconds. That's just an abstraction we use to keep combat tactical and strategic.
Or, put another way: if you're a creature that is benefitting from half-cover, the only way you can then make an attack is if you physically move out of the way of that cover. It may not require you to use up movement speed (if, for example, you're peeking around the corner) but regardless, it's not like a potential assailant would specifically wait for you to make your attack and then retreat before trying to make their own attack; in fact, they'd probably specifically use the moment when you pop out of cover to make their attack.
Just to be clear, I'm not talking in terms of game mechanics; under most conditions, it's not possible for a creature to make an attack during someone else's turn, without using their Reaction or some other clearly specified game mechanic. I'm purely talking in terms of Narrative Diegesis, where "turns" don't exist and everything that characters do in combat is happening (relatively) simultaneously.
But of course that makes it more difficult, hence the mechanical advantage granted by cover of having extra AC, making it harder to successfully land a shot. In this context, the +2AC granted by half-cover is the abstraction meant to explain the difficulty of timing a shot to line up with when you pop out of cover.
This also holds for other kinds of cover: Half Cover abstracts for the fact that, due to being just half cover, maybe part of your body is sticking out beyond the cover outside of your control, whereas with 3/4ths cover, maybe this is no longer an issue. Narratively, how you justify these abstractions is ultimately down to the DM, or for flavor purposes, the players themselves.
So it might seem weird that a 5-ft tree is unable to completely block incoming attacks from hostile creatures, but once you follow how combat is abstracted around cover and the use of obstacles in turn-based combat, I think it's more understandable why these mechanics work the way they do.
Best Answer
Yes.
This does it:
Choosing any other corner gives full cover.