Four of the folds not present in the twelvefold way but introduced in the twentyfold way, rows $5$ and $6$ of the linked table, are defined by the statement that order matters.
However, my understanding is that labeling/de-labeling the elements of the domain and the codomain determine whether or not order matters in the domain and codomain, respectively. These distinctions are already considered in the twelvefold way.
While a physical example might suggest that the relation itself could have an order, i.e., dropping the same balls into the same bins but in a different temporal sequence, in general a relation does not join elements in a particular order.
What concept of order is being used to define these combinatoric categories?
Let’s start with the fairly familiar settings of Row $3$ of the table. The Stirling numbers of the second kind $n\brace k$ count partitions of $[n]$ distinct objects into $k$ non-empty parts; we don’t care about the order of the parts or the order of the objects within each part. If we care about the order of the parts, the number is $k!{n\brace k}$.
Row $5$ is part of what we get when we do care about the order of the objects in each part. Bogart’s example is shelving $n$ books in an empty bookcase with $k$ shelves and then pushing the contents of each shelf to the left. If you imagine shelving the books one at a time, processing them in alphabetical order by author, there are $n$ places to put the first book: you can put it on any shelf. There are $n+1$ places to put the second book, because you can put it on any shelf, and if you put it on the same shelf as the first book you can put it on either side of that book. (Remember, order on the shelf now matters.) Each book that you add to the shelves increases the number of identifiable spots for the next book by $1$, so in the end you have $n^{\overline k}$ possible arrangements (where $n^{\overline k}$ is a rising factorial). The shelves have an inherent order (e.g., from top to bottom), so here we’re partitioning the $n$ books into an ordered collection of $k$ ordered subsets, any of which may be empty.
If instead we simply divide the books into $k$ stacks scattered around the room, allowing any of the stack so be empty, but we care about the order of the books within each stack, the count is different. The Lah number $L(n,i)$ is the number of ways to divide $n$ books into $i$ non-empty linearly ordered subsets, and we allow any number of non-empty stacks from $1$ through $k$, so in this case the number of arrangements is $\sum_{i=1}^kL(n,i)$.
What distinguishes the partitions in Row $5$ from those in Row $3$ is that we now care about the order of the objects within each part. To use your example, if we imagine putting balls into bins, not only are the individual balls identifiable, so that it matters which balls are in which bins, but we also care about the order in which the balls in each bin were placed there. One might imagine the bins as cylinders just big enough in diameter to accommodate a ball, so that the balls in a bin end up sorted from bottom to top in the order in which they were placed in the bin, and different orders are counted as different arrangements of the balls.
Row $6$ is the same thing, except that we have to have at least one book on each of the $k$ shelves or in each of the $k$ stacks, and the reasoning that leads to $n^{\underline k}k^{\underline{n-k}}$ arrangements of books on the shelves and $L(n,k)$ ways to distribute the books into $k$ stacks is similar.
In short, we’re not just counting the ways to divide $n$ distinct objects into parts of some kind: we’re counting the number of ways to divide them into linearly ordered parts. Because the objects are distinct, a part that has $\ell$ elements can be linearly ordered in $\ell!$ different ways, and different orders are counted as distinct arrangements. There need not be any natural or intrinsic order of the objects involved: all that matters is that we can distinguish the $\ell!$ different linear orderings of $\ell$ objects.