The problem lies in the first instance in using the same name for different constructs.
Suppose that you devise an intelligence test, and I do too, and we both call our test results "intelligence": for short, and because we are not innocent of desires to become slightly famous for our work and need a sales pitch, or even wish to make some money out of our tests.
Nevertheless there is no name magic that makes either test -- either measure -- the single, indisputable, incorrigible measure of intelligence.
We should not expect that even rankings will coincide if the two tests are administered to a group of people.
So too with skewness.
The first measure is moment-based skewness, made famous by Karl Pearson, even though he used other measures too, and T.N. Thiele used the same idea earlier. The second measure also goes back to Pearson, although the factor of 3 is just ad hoc to make results comparable with yet another measure he liked, based on mean and mode.
Now, and for some years past, just using (mean $-$ median) / SD looks simpler. And that is bounded in a simple way that is repeatedly rediscovered: $-1 \le $ (mean $-$ median) / SD $\le 1$. The same bounds don't apply to moment-based skewness, although it is bounded according to sample size. For more on that story, see e.g. this paper
Clearly mean being equal to median makes the second measure zero, but that in itself doesn't make any distribution symmetrical. It is easy to find non-pathological distributions for which mean is equal to median, but there is also asymmetry.
A little harder, but it is possible to find asymmetric distributions for which the moment-based measure is zero.
Different criteria of skewness won't necessarily agree. Each is a reduction of information to a scalar, and different reductions ignore different details.
To get back to the beginning: We often write statements like
skewness = whatever recipe appeals
but the equals sign here is rhetorical. What we should be saying is captured better by a notation that I believe has its origins in Algol's use of
:=, and the need to distinguish assignments from tests of equality.
whatever recipe appeals =: skewness
What comes first is the recipe, and any names we use are just conventional. Call the beast an aardvark and it still behaves in the same way.
Here definition is not claiming an unequivocal grasp of a concept, but just declaring a name that is convenient, just as in some very elementary algebra such as
$x = 42$
we are clearly not defining the essence of $x$ but just setting out notation useful for the current purpose.