Help in understanding exactly how lattices used as one way functions for hashing
I'm also afraid you couldn't understand this as D.W., but let us start.
I sometimes cannot understand your questions. Please restate them, if possible.

The definition of the Ajtai hash functions

Let $n$, $m$, and $q$ be positive integers.
Let $R = \mathbb{Z}_q$ be the quotient ring of integers modulo $q$.
Let us define a function, which maps a vector in $D^m \subset R^m$ to $R^n$ and is indexed by a matrix $\boldsymbol{A}  \in R^{n \times m}$ as follows:

$f_\boldsymbol{A}(\vec{x}) = \boldsymbol{A} \vec{x} \bmod{q} = \sum_{i=1}^{m} x_i \vec{a}_i \bmod{q}$.

The Ajtai hash functions is defined as a family of the above hash functions;

$\mathcal{F}_{n,m,q,D} = \{f_{\boldsymbol{A}}:D^m \to R^n \mid \boldsymbol{A} \in R^{n \times m}\}$.

You can obtain your function by setting $n=1$, $m=2$, $q=5$, and $D = R$;
then set $\vec{a}_1 = 14 \ (= 4)$ and $\vec{a}_2 = 13 \ (= 3)$.

For example, you can set $n = 256$, $q = 4097$, $m = 4096$, and $D = \{0,1\}$.
The input of the function is a bit string of length $4096$
 and the output of the function on such parameters is $256$-dimensional vectors whose coefficients are in $\mathbb{Z}_q$.

Is it shrinking?

Good parameters makes the above functions shrinking.
Correctly speaking, if the size of domain is larger than the size of range, that is, $|D|^m &gt; |R|^n$, the function is (obviously) shrinking.

A possible choice of parameters is $D = \{0,1\}$ and $m &gt; n \log_2(q)$, which often appears in the context. 
Another possible choice is $D = \{-1,1\}$ or $D = \{-1,0,1\}$.

Is it hard to find a collision? What is the relation between a collision and a short vector?


  How exactly is this linked to the shortest vector problem or short integer solution if they are not the same thing?


If $D = R$, it is too easy to find the collision is function by the power of the linear algebra. You can use Gauss's elimination if $R$ is a field.
But, if $D \subset R$ and the size of $D$ is small, it becomes hard.
Roughly speaking, the number of collisions depends on the ration $|D|^m/|R|^n$.

In 1996, Ajtai shows that, for good parameters, if there exists an inverter of his hash functions, then there exists an algorithm finding a short vector from any $n$-dimensional lattice.
Goldreich, Goldwasser, and Halevi pointed out in the note you refer that, if there exists an algorithm finding a collision of the Ajtai hash functions, then there exists an algorithm finding a short vector from any $n$-dimensional lattice.
The reductions are based on the fact that the sum of short vectors is (relatively) short.

How should we interpret the above in lattice words?

You would skip this part.

We define a lattice related to a matrix $\boldsymbol{A}\in R^{n \times m}$ as

$\Lambda = \Lambda_q^{\perp}(\boldsymbol{A}) = \{\vec{e} \in \mathbb{Z}^m \mid \boldsymbol{A} \vec{e} \equiv \vec{0} \bmod{q}\}$.

This $\Lambda$ is an $m$-dimensional lattice.
(You can check this is a lattice by verifying that for any two vectors $\vec{e}, \vec{e}' \in \Lambda$, the sum of them lies in $\Lambda$ again.
 You can check this is $m$-dimensional by verifying that any $q \vec{u}_i$ is in $\Lambda$, where $\vec{u}_i$ is a unit vector).

This lattice defines a map from $\mathbb{Z}^m$ to a quotient group $\mathbb{Z}^m/\Lambda$, which is isomorphic to $R^n$.
The map is a hash function.

Conversely, a collision $\vec{e} \neq \vec{e}'$ for $f_{\boldsymbol{A}}$ implies a short non-zero vector $\vec{e}-\vec{e}'$ in $\Lambda_q^{\perp}(\boldsymbol{A})$.