Given an inner product space $V$, you can imagine that there are two different copies of $V$, say $V_1$ and $V_2$, in which each vector $v\in V$ corresponds to a bra $\langle v|\in V_1$ and a ket $|v\rangle\in V_2$. To multiply a bra and a ket together, $\langle v|$ times $|w\rangle$ will by definition be $\langle v,w\rangle$ via the inner product.
Another way to think about this is as $V$ and its Hilbert space dual $V^*$ being identified together; each vector $v\in V$ is afforded the covector $v^*$ which is the linear mapping $v^*(w):=\langle w,v\rangle$ afforded by the given inner product. In this setting the covectors / dual vectors / linear functionals $v^*$ are denoted as bras $\langle v|$ and the usual vectors as kets $|v\rangle$, & multiplication is evaluation $\langle v||w\rangle=v^*(w)=\langle w,v\rangle$.
The reason for $v^*(w):=\langle w,v\rangle$ having $v$ in the second argument is so that each bra is a complex-linear functional of the argument $w$. This is related to a Hilbert space $V$ and its dual $V^*$ being anti-isomorphic; see Riesz representation theorem.
Here is another place where the notation is important: If $\lim_{x \to a} f(x) = b$ and $\lim_{u \to b} g(u) = c$, does it follow that $\lim_{x \to a} g(f(x)) = c$? Of course, the answer is no, but why? Here's an answer using the new notation: Let $u = f(x)$ and $y = g(u) = g(f(x))$. Then we have: as $x \to a^\ne$, $u \to b$, and as $u \to b^\ne$, $y \to c$. But the mismatch between $u \to b$ and $u \to b^\ne$ means that these cannot be combined to conclude that as $x \to a^\ne$, $y \to c$. If $g$ is continuous at $b$, then we can say that as $u \to b$ (no superscript), $y \to c$, and then we can conclude that as $x \to a^\ne$, $y \to c$.
Most calculus books have a theorem about limits of sums, differences, products, and quotients, but no systematic way to work out limits of compositions. The reason, in my view, is that they don't have the notation necessary for talking about limits of compositions. This new notation fills this gap. The new notation is used in a number of places throughout the book. (Similar notation even comes up in the proof of the fundamental theorem of calculus.)
The use of a superscript is consistent with other standard notations that use a superscript to indicate how a variable approaches a limit, such as $x \to a^+$.
In response to the second paragraph of the original post: I think the comments in this paragraph apply to a book like Spivak, but not my book. Spivak covers a lot of topics that belong in an analysis course, and as a result there are lots of standard calculus topics that he doesn't cover. But that's not true of my book, which is really a calculus book and not an analysis book. The arrow notation I have introduced is no more complicated or difficult to use than standard arrow notation, but it makes informal (non $\epsilon$-$\delta$) reasoning about limits more reliable. Isn't that what good calculus notation should do?
Best Answer
The name for what you're after is order statistic.
If you have a sample $X_1, \dots, X_n$ the conventional notation for the $i$th order statistic is $X_{(i)}$, so the second highest value in the sample would be $X_{(n-1)}$ as you suggested.