I don't disagree with you that what people experience as agency isn't what they think it is, but seeing as that's what most perceive free will as being, using a different definition of free will and saying they have it will lead most to think you believe they have that mystical type of agency. I understand the motivation behind using the same term to retain some of the connotation, but it doesn't sit right with me; it feels like I'd be playing a word-game meant to trick myself or others, when I find the argument that we have as much "freedom" as we could want or need stands on it's own without the redefinition. Perhaps it helps some combat a sense of nihilistic depression that stems from their deterministic beliefs, but I can't relate
Re: Libertarian Free Will - I'm not sure whether your interpretation of it is wrong, the definition I found on Google is wrong, or I'm misinterpreting something, but my impression of it is that it's closer to the definition of free will I gave than what you seem to be describing:
EDIT: Reading back I suppose you meant philosophers refer to most people's intuition as libertarian free will. In my defense the phrasing was somewhat ambiguous!