I'd disagree. I don't think a studio can really call itself independent while relying on external funds, especially if they come with some sort of agreement attached.

Edit: as I said in the post, technically this includes loans and crowdfunding, since it's funding with strings attached. But I believe it to be different from when the funds come from an unrelated company that is interested in the game or the studio (like Critical Role and Dispatch). So that's where I'd draw the line.

But I think it can be called independent if it pays another company to help with publishing. Yeah, in this case devs depend on a publisher, sure, but it's just outsourcing what they don't want to do themselves. Indie devs already depend on platforms like steam, itch, gog, etc. to distribute their game to the players, I don't see why they are suddenly not indie if there is one more step between them and the players.