Put simply where we might apply an anthropological theory to art made outside the West - where Science once drew a line between the Primitive and the Civilised - we should apply the same criteria to our own art as we do to the Other. [While we address the Colonialist constructions of Primitive-Civilised dichotomies...]
Accepting or co-opting art made by non-Western cultures into the Art World based on an aesthetic criteria, or a Colonial one - or both - does not treat the work of non-Western culture in the same way as it does its own (and therefore its people). For example, those cultures might not have an Institutional theory of art as we do (which is the theory of the Art World). So it isn't art in the same way as our art is. While Gell might be redressing the prejudices of Colonialism, his theory on art is as much about looking at ourselves with the scientific lens we applied to other cultures in the history of anthropology, to understand art as much as anything.
Where an Aesthetic theory recognises Art by what it looks like, an Institutional theory considers its function within the discourse of Art History. It allows for art that might not be recognised as such on the surface level, rather by its text, eg Conceptual art. Which sounds reasonable enough, as that allows for the dialectical nature of said Art-History, by accepting new things that might not have previously looked like art - and eventually redefining the Aesthetic theory at a later time (where an aesthetic theory is reified, essentialist and even reactionary).
The Institutional theory is sociological and categorises art based on a collective acceptance of something as an art-object. There is no essential definition of art because art can exist in different social paradigms. (This is different to Historical Art, that which was once categorised as Art, or even posthumously categorised as art, such as is the makeup of Museum collections aka artefacts). Because we are talking about what art is now. The categorisation of Institutional Art is helpfully informed by its place in the historical discourse. Make a work that refers to Duchamp's Fountain and it will be considered for acceptance as Art. Put it - or anything - in a gallery and it is considered Art because it is discursive. Attempt to contribute to Art History (discourse) and you have Art, the Institution being The West, not just art galleries. The Institution is History.
Anthropological art theory looks outside of this discourse in an attempt to not just dissolve the Western Colonial Story, but to understand ourselves, primarily. Gell was an anthropologist after all, and interested in the human subject. Art being critical to the study of that subject. His third definition builds on the second, and draws on his studies of non-Western cultures - as has anthropology historically. In his theory he gives the art-object Agency, and posits it with agency and person-hood within a social framework. He doesn't essentialise art as this, as it is a theory of art as well as people. But understanding an art-object as a functioning, alive thing - equal to that of people - within society is a totally new way for me to understand art and I hope to continually understand and apply his theory and ideas in my practice, both academically and in the studio for the foreseeable future.
Alfred Gell died at the age of 51, before his book Art and Agency was published, but his ideas on art and anthropology and indeed the Anthropology of Art have continued to feel relevant and have inspired me greatly so I thank him for them.