In the lesson linked to in your post, it is explained that propre can also mean "own", depending on the position of the adjective.

Je dors dans mon propre lit. -- I sleep in my own bed.
Je dors dans mon lit propre. -- I sleep in my clean bed.

In the case at hand, propre is separate from the noun. In this case the meaning is determined by context. A "clean" culinary culture doesn't make much sense, so it's one's "own" culinary culture, or -- better yet -- one's specific culinary culture.