In simple terms what is the difference between Fisher and Pearson kurtosis definitions?

Question

I have read that there is a difference in kurtosis of normal distribution, but I don't quite get it intuitively. Why do two definitions exist?

Answer 1

From comment:

The fourth standardized central moment $\operatorname{E}\left[\left(\frac{X - \mu}{\sigma}\right)^4\right]$ is bounded below by $1$ and is $3$ for a normal distribution. I would call this the kurtosis.

But since a normal distribution is "mesokurtic", you can subtract $3$ to give what I would call the excess kurtosis and say a distribution with negative excess kurtosis is "platykurtic" and one with positive of infinite excess kurtosis is "leptokurtic".

Excess kurtosis is more convenient working with cumulants; using kurtosis may be more convenient with multivariate distributions.