“Adjacent” means what it does in English—directly bordering one another. Two squares are adjacent if they share a corner or edge. Two creatures are adjacent if their spaces include two adjacent squares. This is the same as the squares that are within five feet of each other, yes, since the grid is 5-ft. squares. This does not change with creature size—adjacent always means the sides must be touching on at least one corner or edge.
But all of this just comes from what the word adjacent means. That parenthetical might well be the only explicit description of it, but it can’t be “the official definition” or anything, not in a parenthetical on one very specific topic. Rather the word isn’t defined by the game rules at all, and “the official definition” is instead found in any dictionary.
Anyway, so, no, Bodyguard does not change its range with reach (the number of adjacent squares does increase with space, though, since there are 8 squares around a 1×1 space and 12 around a 2×2 space). That said, I tend to agree with you that some uses of things affecting adjacent squares would make more sense affecting the natural reach for a given size (i.e. not subject to any bonuses to reach that a creature may have on top of their size). But that kind of ruling has to be made by the GM. If you’re not the GM, you’ll have to just ask them about it.
If you are the GM, when deciding if a given feature should use natural reach instead of adjacent squares, to be fair and honest you should decide that ahead of time and let players know about it. Also, keep in mind that Tiny and smaller creatures have a natural reach of 0 feet—should they only be able to use this feature while sharing a space with something? (Maybe! But it’s something to think about when making the judgment, since you should be consistent.)
In the case of “it just came up in the middle of a fight and I only now realize I want to modify the rules,” my usual approach is to make the houserule immediately only if it benefits the PCs. If it would harm the PCs, I make the houserule after the fight is over, for any future fights. In both cases, I let the players know, and if a player decides that the change makes them wish they had taken that feat or whatever, I work out a way to let that happen. (There can be exceptions to either in very unusual circumstances—maybe a boss fight would be completely neutered if I don’t act, and that’s no fun for the players either—but I can’t remember the last time I did that.)