I disagree with you and the majority of commenters. I haven't read past page 1 but this is my understanding:
Vegetarian - does not eat meat or fish. Can consume animal products, such as eggs, milk, extracts, butter, fat, and so on.
Vegan - does not eat meat, fish, or animal products. No animal fats, no honey, no eggs, and so on, or anything that is produced from an animal, such as rennet.
Obviously as human beings people are welcome to twist and change the definitions of these ideologies however they please, but traditionally vegetarians CAN consume animal products. It is vegans who CANNOT.
So I don't think what the chef has said here is wrong at all - cheese is vegetarian, even those containing rennet, as they do not contain MEAT or FISH and and just contain animal products - which vegetarians are allowed and do eat, by traditional definition.
If vegetarians can't consume animal products, what is the distinction between vegetarianism and veganism?