Was the word "culture" first used as a medical term, or first used as a social term?
According to the Oxford English Dictionary (and if anyone knows, it does) the "social", or "cultural" definition of the word "culture" is much older than the biological, medical definition.
The "social" definition of culture comes from a farming term, meaning in which "culture" means "The action or practice of cultivating the soil; tillage". This branches off into
The cultivation or development of the mind, faculties, manners, etc.; improvement by education and training
and the first recorded use of the word in this sense is around 1510.
This in turn seems to lead to the definition of culture as
Refinement of mind, taste, and manners; artistic and intellectual development. Hence: the arts and other manifestations of human intellectual achievement regarded collectively
which is, I think, what you mean. First recorded use of the word in this sense is 1678.
The biological definition comes much later: defined as
The artificial propagation and growing of microorganisms, or of plant and animal cells, tissues, etc., in liquid or solid nutrient media in vitro. Freq. attrib.
the first recorded use is 1880.