I understand how the question arrived at the conditional probabilities, however couldn't the question also be interpreted this way?

Since the question never stated that the 2 dice were different, what if we assume they were not distinct and also we can assume that the 2 dice is thrown at the same time, thus $\displaystyle P(S = 4 | M=3) = \frac{P( S=4 \cap M = 3)}{P(M=3)}$

Now to have a sum of 4 and a max of 3, all we need is to roll a 1 and a 3, there is no order here since we are throwing the dice at the same time and the dice are NOT distinct. Thus the numerator becomes 1/6 * 1/6

To get a maximum of 3, all we need is to roll a 1 and a 3 or 2 and a 3 or 3 and a 3 (again there is no order since we are throwing the dice at the same time)