I am not sure I get your understanding of peakedness and heaviness. Kurtosis means "Excess" in German, so it describes the "head" or "peak" of a distribution, describing whether it is very wide or very narrow. Wikipedia states that the "peakedness" is actually described by the "kurtosis", whereas peakedness does not to appear to be a real word and you should use the term "Kurtosis".

So I think you might have gotten everything right, the head is the Kurtosis, The "heaviness" of the tail might be the Skewness":

Here is how you find it:

$$ a_3 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^3}{N * s^3_x} $$

with s as the standard deviation for x.

The values indicate:

Negative Skew: $$ a_3 < 0 $$

Positive Skew: $$ a_3 > 0 $$

No Skew $$ a_3 = 0 $$

You can get a value for the kurtosis with: $$ a_4 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^4}{N * s^4_x} $$

The values indicate:

Wide Peak: $$ a_4 < 0 $$

Narrow Peak: $$ a_4 > 0 $$

Normal Peak, so Normal Distribution $$ a_4 = 0 $$

Did that help?

I am not sure I get your understanding of peakedness and heaviness. Kurtosis means "Excess" in German, so it describes the "head" or "peak" of a distribution, describing whether it is very wide or very narrow. Wikipedia states that the "peakedness" is actually described by the "kurtosis", whereas peakedness does not to appear to be a real word and you should use the term "Kurtosis".

So I think you might have gotten everything right, the head is the Kurtosis, The "heaviness" of the tail might be the Skewness":

Here is how you find it:

$$ a_3 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^3}{N * s^3_x} $$

with s as the standard deviation for x.

The values indicate:

Negative Skew: $$ a_3 < 0 $$

Positive Skew: $$ a_3 > 0 $$

No Skew $$ a_3 = 0 $$

You can get a value for the kurtosis with: $$ a_4 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^4}{N * s^4_x} $$

The values indicate:

Platycurtic: $$ a_4 < 3 $$

Leptocurtic: $$ a_4 > 3 $$

Normal: $$ a_4 = 3.0 $$

Did that help?