It has to be more nuanced than “self-published”

It doesn’t need to. Definining it as “self-publishing” is enough.

otherwise everything EA craps out is “indie”.

And because of the above, EA games might very well fit the definition, yes.

This clearly shows that maybe we shouldn’t use “indie” to describe good games (or the lack of it to describe bad ones). It should just be used to define “means of publishing”.