What is skewness of a distribution?
I ask it why any particular indices seem indecisive about symmetry, and in some case also about asymmetry.
Skewness is related to the symmetry of a distribution.
Note that I do not write that "skewness measures symmetry" or some such. The specific relationship between symmetry and skewness is a bit complicated.
A symmetric distribution will have zero skewness, for usual definitions of skewness. (Yes, there are multiple ones.) For instance, in Pearson's moment skewness, the third power in the formula implies that probability masses to the left and the right of the mean cancel out.
However, the converse is not true. You can easily create distributions that are not symmetric but whose Pearson's moment skewness is zero - we just need the densities to cancel out. In fact, you can do this for unimodal distributions as well. The same applies to other skewness measures, like Pearson's mode skewness or median skewness.
However, for practical purposes zero skewness is usually treated as equivalent to symmetry, and unless you purposely create a pathological example, a zero skew distribution will usually be close enough to symmetry that you will be fine.