This is how that link said to list them:
“If your employee compensation includes stock options, report only the net
value (after taxes, commissions, and fees) of your vested stock options.
Report the value of these vested stock options as if they were exercised and
sold at market value on the exact day you complete the FAFSA form.”
I think I saw the same link. I have not yet seen any articles or resources that specifically differentiates between vested options held at a private vs public company, but my guess is that we ARE supposed to to include the private options in the assets reported on FAFSA, even if there are some good reasons why this may not make sense.