What Are the Differences in Four Different Types of Third Order Image Moments (Skewness)?
It it based on the definition of Central Moments on 2D grid given in the Wikipedia Page.

$$ {\mu}_{pq} = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} (x - \bar{x})^p(y - \bar{y})^q f(x,y) \, dx \, dy $$

Since we're dealing with 2 random variables, a multiplication which the powers sum to 3 can be achieved in 4 different methods.

It's up to you to decide how to calculate the third moment.

Usually I'd use the Rotation Invariant Moments.
You can find the definition on the same page.
Enforcing the Rotation (And Scale) Invariance yields single definition.