I did some checking on the word "self", and found an interesting link on autism.
http://iautistic.com/autism-theory-of-mind-revisited.php
The author of the article is an autistic, and he describes autistics as people who have a self that functions differently than non-autistics.
What implications, if any, does this have on our 'everyday' notion of self?
I also found this in a wikipedia article on 'cognition':
"The term "cognition" is also used in a wider sense to mean the act of knowing or knowledge, and may be interpreted in a social or cultural sense to describe the emergent development of knowledge and concepts within a group that culminate in both thought and action."
I don't know if it's relevant, but it seems to me that this is about "cultural dimensions" of cognition.
So yes, "self" has a cultural dimension.