To zoom in here, because this is a point of disagreement between us, I'm not sure what value would be gained by expanding the definition of protagonist play to something other than focusing on the dramatic needs of the character. There's a distinct difference in focus outlined by this definition, and it's one that's useful because it goes a good way to exploring what a game is about. If we expand this to other things, like, say, just a PC goal, then we've made this distinction useless, and will just need a different term to discuss it. If the goal is to make the term protagonism useless because people don't like the way the term is used, that's a different thing.
So, why do I think the distinction between a character dramatic need and a character goal is important? Because it isolates the focus of play. If play is about my character's dramatic need, then it's focused on the character -- things are framed in terms of these needs and outcomes revolve around them. This makes the character the focus of the game, but doesn't, at all, mean they get a break. If anything, this focus is far more punishing on the character, and more exhausting to the player, than a game where such focus isn't held. I think this makes for a distinct difference in play that's not aided by expanding the definition. And it's not an absolute term -- a game doesn't have protagonism or not, there can be a mix, but I also think that this mix is a difficult thing to pull off, because it requires switching focus between a character dramatic need and an NPC dramatic need. That's a good challenge, but it also muddies the focus of the game, and I'm not sure if there's a great deal of use to mixing it up. Perhaps a mostly non-protagonism game can do so with short side-treks of protagonism, but I think the vice-versa would be a bit more jarring to a player. Maybe not, willing to listen to other opinions.
But, and this is key, it's very important to differentiate between a character dramatic need and a character goal. Character goals don't have to have anything at all to do with the character. For example, stop the evil overlord from summoning the demon apocalypse is a great character goal, but it's not a dramatic need. The difference is that this goal doesn't speak to anything at all fundamental about the character, it's just something the character is going to do. I can swap in a different character and have the same goal with no change. Dramatic needs should be special to the character. If I have a dramatic need of "will I be able to resist my alcoholism and support my friends," then this is special to the character -- you can't just swap in a different character and have this remain the same (if you do, I'd question just how invested in playing to dramatics needs are to you). This is a key difference. Likewise, a character goal to research a new spell is fine, but not a dramatic need. It may server a dramatic need, but it isn't one itself. This is a critical difference I think has been glossed a number of times in this thread (general, not specific, you here).
And, of course, you can have multiple dramatic needs from multiple characters. Speaking to the singular is for clarity and simplicity of the post, so I don't have lots of "or that of other characters" floating about.