No, this distinction prevents large studios from co-opting “indie” as a label, which people support because of that artistic discretion, and hiding it behind their opaque promises of such independence that no one can verify. You cannot trust a dev hasn’t been influenced by a publisher when they’re present, so the only way to ensure that is to not have a publisher present.

I don’t know that movie, but I do know actual indie devs who use e.g. Patreon for funding. It’s not about not having money, it’s about who your money comes from, and whether there can be hidden stipulations on it. With publishers, there always are.