Correct, the taxes are not added to the basis. I believe Thomas was saying that you did pay taxes and that word got stuck when he later meant to say exercised the option or fees.
The link you sent does a great job explaining. If you understand the example, you are set! You have the cost basis part that you paid plus the compensation from your w2 along with any fees paid- not taxes, commissions paid to exercise the options, if any.
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