Name of theory stating "worldview affects language structure"

I'm not sure there's an accepted term for this. I think this could still be conceived of as a type of 'linguistic relativity', though. Especially, since there's always an interplay between 'reality', culture, and language even in cases of the Sapir-Whorf style of relativity.

Daniel Everett has been proposing a theory of how the Pirahã's evidentiality (and other) features are influenced by their cultural insistence on witness testimony for topics of conversation. He discusses it both in the context of and in contrast to linguistic relativity but I don't think he has a specific label for this position. (See e.g. this paper and his book).