Probably, that site knows what a "two-sided test" means whereas, judging by your description of simulation for the skewness, you do not. The simplest change to your procedure would be to use the absolute value of the calculated skewness, since that is the test statistic for a two-sided test in this case. On the webpage, "alpha" is the total area of the two tails, not just one tail.
You also said "Step 2a: for the kurtosis I need 2 critical values, but for the skewness do I really need 2 critical values?". You do need two critical values for the raw skewness, but for symmetric distributions you know that these are related in a simple way. If you were working out a test of skewness for some non-symmetric distribution, as is certainly possible, there would be non-symmetric lower and upper limits for a two-sided test.