Sum of dice adding to less than $5$ I have seen some places saying that the sum of $2$ dice being thrown being less than $5$ can occur for the cases

$$(1,1),(2,2),(2,1),(3,1),$$

giving a probability of $\frac4{36}$ for the event.

However, I have also seen people counting the numbers twice, as there are two different ways to throw the dice, giving the cases

$$(1,1),(1,1),(2,2),(2,2),(2,1),(1,2),(3,1),(1,3),$$

and a probability of $\frac8{36}$.

Which way is correct?
Neither is correct.  The second approach is the correct one – you list all the ordered possibilities. Nevertheless, you have listed $(1,1)$ and $(2,2)$ twice.
Neither.

There is a single way to roll a sum of 2: roll $(1,1)$.

There are two ways to roll a sum of 3 and two ways to roll a sum of 4, as you pointed out, where the numbers are different. Notice that $(2,2)$ also gives a sum of 4.

So the total probability is $6/36$.

One nice way to see this visually is to make a table, one with the value of the first die along the columns and the value of the second die along the rows, and record their sum at each cell in the table. The table will have $6 \times 6 = 36$ cells and you'll see that the diagonal rolls $(1,1)$, $(2,2)$, etc. all only appear once, while all the others appear twice.