Understanding the definition of Sylow $p$-subgroups

As I mentioned in comments, I think the issue here is actually that we misname "maximal subgroup".

Of course, given a partially ordered set $P$, a maximal element (or $P$) is an element $p\in P$ with the property that for all $x\in P$, if $p\leq x$ then $p=x$.

If we have a class $\mathfrak{X}$ of subgroups, partially ordered by inclusion, we often talk about "maximal $\mathfrak{X}$-subgroups" (also "minimal $\mathfrak{X}$-groups", though that is less common). Thus, "maximal $p$-subgroup" means "maximal among the $p$-subgroups".

We say "maximal subgroup" (and "maximal normal subgroup"), but we really mean "maximal proper subgroup" (that is, the class $\mathfrak{X}$ is the class of proper subgroups, not the class of all subgroups). (And we likewise really mean "maximal proper normal subgroup").

So we should say:

A maximal proper subgroup of $G$ is a proper subgroup $H$ of $G$ that is not properly contained in any proper subgroup of $G$.

If we used that terminology and definition, you would likely have no problems. But it is an unfortunate fact that nobody says "maximal proper subgroup", we just say "maximal subgroup" and elide the "proper" clause, on the (flimsy) excuse that of course we must mean proper subgroup, as otherwise the concept would just be "the group $G$" and it would be silly to introduce such a nomenclature for the whole group.