Thank you for all the replies. To clarify, cheese is not really going to get me killed, so I ended up eating some anyway. Now I understand the rules.
How can I order a "tiny-amount-OK-but-not-100%-cheese" meal? I don't think it is listed as one of the options for requests. Besides, it probably looks stupid to tell FAs on every flight up-front that "hi, I sit in XX, and if one of the meal options is cheese, please make sure that I don't get it."