Ebyan Alvarez-Buylla
The Punk Rock analogy is probably the best fit here, especially since it is so loose and equally up for debate.

The most inclusive way of looking at "indie" gamedev goes something like:

Small team OR self-funded/not backed by a major publisher OR adhering to the established indie aesthetic

While the less inclusive, more snobby definition is something like:

Small team AND self-funded/not backed my a major publisher AND adhering to the established indie aesthetic

As you point out, "small team" or "self-funded/not backed by a major publisher" or "established indie aesthetic" all cover wide spectra of subjectivity and in some sense don't get us any closer to a consensus. However, I will say that the latter of those three resonates more with me as a full-time non-indie, part-time indie, since it tends to be what makes the most difference in how the game feels. Really, "philosophy" is a better term than "aesthetic", because we're not talking entirely (or perhaps at all?) visually, but rather hinting at the more niche/experimental results of having to make due with a small team and no money, an aesthetic that can be to some extent emulated with a larger team and plenty of funding.

Visually, one way I like to think of the gamedev space is as a 2-dimensional spectrum: in one axis you have experimental->established, and on the other you have niche->casual. "Indie" games tend to fall in the experimental/niche quadrant, though some have been known to peek into the casual space. AAA games hover around the established axis, taking care not to become too complex as to teeter too deep into the niche extremes and keeping just the right amount of explosions and gratuitous violence to keep them from becoming casual.

In many ways I am quite pleased that these sorts of arguments are beginning to be had by the wider game playing public much in the same way music has enjoyed the 'what is "punk"' argument for decades. Even if many of us gamedevs are as keen on limiting labels as we are on being harassed over a $2.00 price tag on a mobile game, this sort of snobbery is indicative of a healthy, growing game development culture.