Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$

There are, as you pointed out, four types.

For the three distinct roots, it is $\dfrac{p(p-1)(p-2)}{3!}$. Whatever order you pick the three distinct roots, you get the same cubic.

For the two identical, and one different, it is $p(p-1)$. (There should not have been division by $2$: a double root of $a$ and a single of $b$ is different from single $a$, double $b$.)

For all identical, it is of course $p$.

And you are right about the quadratic times linear, it is $\dfrac{p^2(p-1)}{2}$.

Calculate. We get the right answer.

The general formula is obtained by G. J. Simmons in The Number of Irreducible Polynomials of Degree $n$ Over $GF(p)$.