$\overrightarrow{XY}$ has an initial point, but $(x,y)$ doesn't, despite $\overrightarrow{XY}=(x,y)$: how to reconcile? Allow me to illustrate with an example: define two coordinates in $\mathbb{R}^2$:


*

*$A=(1,1)$,

*$B=(2,4)$.


We now have three observations:


*

*$\overrightarrow{AB}$ has a initial point, namely $A$.

*$\overrightarrow{AB}=(1,3)$.

*$(1,3)$ does not have an initial point (and, if it did, it certainly would not be $A$).


Question: How can we reconcile these three, seemingly incompatible, observations?
The inconsistency comes about since $$\overrightarrow{AB}=\overrightarrow{(1,1)(2,4)}$$ consists of four real numbers, rather than two.  This makes me think vectors of the form $\overrightarrow{XY}$ are inherently different to vectors of the form $(x,y)$.
One way I can think of that might work is to treat equality among vectors as an equivalence relation, with equivalence classes $$\{\overrightarrow{XY}:Y-X=(x,y)\}$$ denoted by the shorthand $(x,y)$, analogous to how $a \in \mathbb{Z}_n$ represents an equivalence class of integers.
 A: 
One way I can think of that might work is to treat equality among vectors as an equivalence relation

I think that's the standard solution to this problem! Here's one reference that uses it.
By analogy, the fraction $\frac12$ has a numerator of $1$, and the fraction $\frac24$ has a numerator of $2$, but as rational numbers, $\frac12=\frac24$. This makes sense, because a fraction is an ordered pair of integers, while a rational number is an equivalence class of fractions. It's common to abuse notation here, so when we write "$\frac12$" we might mean either the fraction or the rational number. The only way to tell is by the context.
A: It seems to me like you have a few competing notions of vector in the question. Perhaps it would do you good to distinguish between positions and displacements, that is to say, places and differences between places.
The trouble being that positions are often identified by a displacement from some fixed reference point (which becomes the origin), and displacements are sometimes identified by the position you end up with if you add them to a reference point. So in terms of the actual data that we use to express positions or displacements, they are frequently interchangeable, even though conceptually they are quite different things.
So, in your examples, I would say that $\overrightarrow{AB} = (1,3)$ is a problematic statement because the left and right hand sides are different sorts of things. It's true if you're informally identifying mathematical objects with their representations (which is, to be fair, almost always) but not really a well-formed statement if you're being pedantic about the meanings of ideas.
It may also be the case that "$\overrightarrow{AB}$ has an initial point, namely $A$" is more of a statement about how $\overrightarrow{AB}$ came to be, or was constructed, a statement about its progeny, than about actual properties of the vector itself. Much like I might say I'm British even though if you slice me open you don't find atoms of Britishium inside. Certainly the way I've studied vectors in my undergraduate, I've never considered them to come packaged with initial or final points, they are just sort of "arrows" which may be rooted wherever they like.
Further reading: an affine space is a mathematical structure where there are points and vectors, and you can subtract points to get vectors, add a vector to a point to get a point, and add or scale vectors, but you can't add or scale points. I've had affine spaces described to me as what happens when you start with a vector space and forget where the origin is.
Further thinking: I find that temperatures provide an interesting analogy. Once upon a time, all we knew about temperatures was that various liquids expanded and contracted in consistent ways in response to heat. They followed a roughly consistent temperature/volume curve until they froze, at which point you had to use a different liquid with a lower freezing point. At the time it didn't make sense to talk about one temperature being "twice as hot" as another, only "this temperature is as much hotter than that one, as that one is than the other" – we could only measure differences. We invented reference points so that we could express these differences with numbers, but of course it is nonsense to say that 70 degrees Fahrenheit is twice as hot as 35 degrees Fahrenheit, since they are roughly 21 and 1 degrees Celsius respectively. However, it does make sense to say that the difference between 10 and 20 degrees is twice the difference between 32 and 37.
Of course, subsequently we discovered absolute zero, and now temperatures may be regarded as absolute – 100 Kelvin really represents twice as much heat energy than 50 Kelvin. (Sort of. Don't eat me, physicists.) In a sense, we have discovered the origin of our affine space, turning it into a vector space.
