Identifying the C3, C4, S4, and S6 symmetry operations in the Oh point group
At first I thought the character table was counting both positive and negative rotations, but then the order for the $C_2'$ and $C_2$ rotations should be doubled if that were the case ($12C_2'$ and $6C_2$).


You were very close. The missing symmetry operations are indeed ‘positive and negative’ but only those that lead to non-identical transformations. In a $C_2$ axis, it does not matter whether you turn clockwise or counter-clockwise. Consider the following mock-up of the central planar square of an octahedron with the ligands labelled A, B, C and D:

 A
BMD                      (1)
 C


If you apply $C_2$, you will arrive at the same orientation (2), regardless of which direction you rotate in:

 C
DMB                      (2)
 A


However, when rotating by $C_4$ clockwise and counterclockwise actually lead to two different results (3) and (4):

 B
CMA                      (3)
 D

 D
AMC                      (4)
 B


These two are not identical transformations. In fact, they have distinct symbols; depending on your direction of rotation one is $C_4$ and the other is $C_4^3$. You can also interpret them as rotation by $90^\circ$ and rotation by $3 \times 90^\circ = 270^\circ$. $C_2$ and $C_4^2$ correspond to the same transformation as well.

The same case can be made for $C_3$, the reverse rotation is $C_3^2$.

The $S_n$ cases are not quite as simple. Well, actually $S_4$ is ($S_4$ and $S_4^3$) but that’s a coincidence. $S_6$ shows this more clearly. Remember that $$S_6 = C_6 \oplus \sigma_\mathrm{h}$$

Therefore, performing $S_6$ twice is equivalent to $C_3$:

$$S_6 \oplus S_6 = S_6^2 = C_6 \oplus C_6 \oplus \sigma_\mathrm{h} \oplus \sigma_\mathrm{h} = C_3 \oplus E = C_3$$

Thus, $S_6$ applied twice and $S_6^5$ applied twice give $C_3$ and $C_3^2$, respectively, so $S_6^2$ and $S_6^4$ are already taken care of. A last case needs to be made for $S_6^3$: This is identical to $i$ as can be seen below:

$$S_6^3 = S_6 + S_6 + S_6 = C_6 \oplus C_6 \oplus C_6 \oplus \sigma_\mathrm{h} \oplus \sigma_\mathrm{h} \oplus \sigma_\mathrm{h} = C_2 \oplus \sigma_\mathrm{h} = i$$

Thus, while you could imagine up to six different $S_6$ transformations, only two of those ($S_6$ and $S_6^5$) give new transformations not covered elsewhere.