What do I not understand about one-to-one functions? Firstly, a definition:

Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. 

Now the question:

Students often misunderstand the concept of a one-to-one function (mapping). You see, a mapping has a direction associated with it, from $A$ to $B$. It seems reasonable to expect a one-to-one mapping that carries one point $A$ into one point of $B$, in the direction indicated by the arrow. But of course, every mapping of $A$ into $B$ does this, and Definition 1 did not say that at all. With this unfortunate situation in mind, make as good a pedagogical case as you can for calling the functions in Definition 1 two-to-two functions instead.

Definition and question ever so slightly modified from A First Course in Abstract Algebra by Fraleigh.
So this is my thoughts: First of all, despite the fact that Definition 1 did not say this, I thought it could be more or less inferred. As in, there is a formal, and intuitive definition of a derivative, but they more or less mean the same thing. But, I guess if this were considered a two-to-two the best argument I could think of is given $x_1 = x_2 \in A$, and $\phi(x_1):=y_1 = \phi(x_2):=y_2 \in B$, then the "diagram", so to speak looks more or less like:
$$ x_1 \rightarrow y_1 \leftrightarrow y_2 \leftarrow x_2 $$
So there is a kind of a two-to-two symmetry, but to me, it is a very weak (couldn't we just cut out the middle anyway?) argument, and don't believe this is what the question is getting at. Can someone explain to me what most students don't understand?
 A: Re: ``so an injective function is not necessarily a function, in terms of function vs relation? That is misleading.'' 
Your are absolutely correct: a relation may is injective.   A relation R from A to B is injective when $R^{-1}\  o \  R$ is contained in the diagonal id$_A = \{(a,a): a\in A\}$ of $A\times A$ (the graph of the identity function on $A$).
The single-valuedness of a relation, being a partial function, is the dual property: $R$ is single-valued (a partial function, one defined on a subset of $A$) when   $R \ o\  R^{-1}$ is contained in the diagonal id$_B=\{(b,b): b\in B\}$ of $B\times B.$  Duality is in that $R$ is replaced with $R^{-1}.$
Now replace $\subseteq$ with $\supseteq$ in the first definition to arrive at $R^{-1}\ o \ R\supseteq$ id$_A.$  A relation with this property is a correspondence.   A function is a single-valued correspondence (full domain).
The dual notion of the preceding one, $R \ o\  R^{-1}\supseteq$ id$_B,$  defines a surjection.
A: one-to-one, misunderstood: "for every x, there is only one y=f(x)". This is the definition of a function
one-to-one: "if you have two different Xs, their f value is different"
"two-to-two": if you pick two different Xs, and evaluate the function on each, you get two different Ys
A: I agree that one-to-one is confusing, and agree that two-to-two neatly captures the idea. But the existing vocabulary is so entrenched that even recommendations from luminaries like Terence Tao and John Conway won't change it.
I try to convince my students that mathematicians only define things after they understand what they are trying to define. You don't really learn a concept from its definition, you learn it from examples and counterexamples (easy to provide for various kinds of mappings). Once you understand the concept you can make sense of a formal definition, even if it's ugly and the concept isn't well named.
A: Searched and found this comment by Terrence Tao on a blog at this link:
Too long for a comment.

Terence Tao Says:
John H. Conway once observed that the common terminology “one-to-one” for injections is horribly misleading, as it suggests that the definition of an injection is “something that sends one input to one output” – which is instead the definition of a function, not an injective function. (The problem is that “one-to-one” is intended to be in contrast to “many-to-one”, but is often instead viewed by beginners as a contrast to “one-to-many”.) He proposed instead the more accurate terminology “two-to-two” (sending two distinct inputs to two distinct outputs). This hasn’t caught on, however (not surprising though, given the severe alliteration).

A: Most students (I was one of them) thought  incorrectly that one-to-one function means that for any x, f(x) can only have one value. 
Injective seems to be a less confusing word.
A: As the other answers indicate it seems you correctly identified the source of confusion for most students and your proposed solution (two-to-two mapping) is really elegant I think. Nevertheless, since it is not in the answers above, here another way in which the term one-to-one can confuse students (read: confused me as a student).
Living in Holland I had been familiar with the concepts injective and surjective (under their Dutch names injectief and surjectief) for quite some time before starting to read math texts in English. Deducing meanings of words from context and similarities between languages is often easy enough and so the first so many times I encoutered the term one-to-one mapping I read it as [mapping establishing a] one-to-one correspondence. (In other words: a bijection.)
Only after the tenth or so instance of someone explicitly emphasizing that a map was onto in addition to being one-to-one, I figured that it might be a good idea to maybe look up the definitions of these terms...
