TITLE: Why does a bijection from a set to itself deserve the name "Permutation"?
QUESTION [9 upvotes]: Sorry for the long text; this is a nebulous question that has always been in the back of my mind, and I've had trouble putting into a short form.

"Natural" Definition
If someone on the street hears the word "permutation," I think they will naturally assume that a permutation:

Involves rearranging objects
The order in which the objects are written, before and after the permutation is performed, is of crucial importantce (it's really the essence of the permutation)

I would naturally expect a permutation to be an instruction, or an action. For example, I would expect a permutation to look something like
$$\sigma = \text{interchange the first two entries.}$$ 
Then, if we apply $\sigma$ to $(A, B, C)$ we get $(B, C, A);$ if we apply it to $(4, 6, 9)$ we get $(6, 4, 9).$ To me this is a very satisfying (informal) definition of a permutation, because it captures exactly what many people (or at least me) would think a permutation "should be."
Another way to define "permutation" (to me, less satisfactory than the previous, but still more satisfactory than the official definition) could be to just say that "the 3-tuple $(B, A, C)$ is a permutation of $(A, B, C).$" (In fact, I think this is the definition used in elementary Statistics books.)
Percieved Weaknesses of the Official Definition

It makes little sense to "permute" your set of objects. If you have a set of objects $\{4, 6, 8 \},$ and while you are not in the room someone applies a permutation to your set, you will never know; the output of your permutation is still $\{4, 6, 8 \}.$ Even if they only apply the permutation to a subset, you only might be able to tell.
Permutations seem to have nothing to do with the order that your objects are in, either before or after doing the permutation. This, like I mentioned above, seems to violate the whole point of a permutation

I call these weaknesses because they seem to violate the "person on the street" understanding of a permutation, and I know that generally mathematicians try really hard to not distort the meaning of common English words too much.
My Question
Is there really such a big disconnect between the "Natural" and Official definitions of permutations? Even if there is not, and there is a way to tediously link the natural definition with the official definition (which I'm sure there is), why does the Official definition deserve to be called a permutation more than the natural one? Is there a name for the Natural definition? 
Thanks.

REPLY [9 votes]: "A bijective map from a set to itself" does not require the set to be ordered, but when applied to an ordered set, this map acts to reorder the set.
This definition is therefore a generalization of the idea of "reordering an ordered set" to a more general setting.
Often, in mathematics, a name lifts with a generalization.