Free Software Foundation appear to disagree with your definition [1], while OED lists absence of monetary charge as only one of necessary characteristics of "free software" [2]
I don't care and it doesn't matter what the FSF thinks, it's human language and they can't dictate how that works.
I'd assumed that the main difference between the capitalised version and uncapitalised version was being potentially able to trademark the former, so you could get the distinction in free-ness that way. I'm not aware of the FSF actually trademarking the term though, so I could well be wrong on that point.