To me the small meaning of "media" is the physical, external-to-us part of the human artisphere that uses language, orthography, writing, theater, video, etc. to share among us. The large one is everything that is between any two or more (sentient?) beings, or any number of groups of us. I usually tag with the small meaning.

There is an excellent critique of the small meaning as an ongoing conversation in anarchism - about media coming between people, alienating us from each other, and from the rest of the world generally. It's a valuable critique, and understandably one-sided given how much alienation there is, but I think it's more accurate to say that words (and media in general) can be windows or they can be walls.