Wikipedia says that the skewness of the log-normal distribution only depends on the variance of the underlying normal distribution.
Skewness: 
However, from my point of view the skewness increases as the mean of the underlying normal distribution increases (described here).
Can anyone show which point is valid?
I believe the ambiguity comes from how you define skewness. Wikipedia uses $$ \mathrm{E}\left(\frac{X - \mu}{\sigma}\right)^3, \text{ with } \mu=\mathrm{E}X \text{ and } \sigma^2=\mathrm{E}(X-\mu)^2. $$ If you compute this for $X\sim \text{log-normal}(\mu,\sigma^2)$, you arrive at the given formula. During the calculation, the dependency on $\mu$ is normalized out when you divide by $\sigma^3$. To check this, write $X = \exp(\mu+\sigma Z)$ for $Z\sim\text{Normal}(0,1)$ and try calculating the moments.
Alternatively, the third moment, $\mathrm{E}X^3$, and the third central moment, $\mathrm{E}(X-\mathrm{E}X)^3$ certainly do depend on $\mu$. Your cited web page doesn't define skewness, but I assume they are referring to the third central (unnormalized) moment.
