I say "right-wing libertarian" because the term "libertarian" was coined by a French anarchist in the 19th century, and AFAIK, outside the US the term "libertarian" still has that left-wing connotation and meaning. So it's at least useful to clarify by libertarian I mean conservative liberals and not communists / anarchists.
In the US, the conservative or right-wing liberals adopted the term "libertarian" around the mid 20th century to distinguish themselves from social liberals, and since then the term has just come to mean conservative liberal as you mention - I don't know many people in the US aware of other meanings of the term "libertarian".
As an aside, I'm not as willing as you to paint social democrats or social liberals as "an inherently right-wing ideology" because we would get into trouble with the fact that social liberalism represents the majority of what we might call "leftist", particularly on social policies.
This disagreement isn't really about the fact that social liberalism has right-wing positions, we agree that on the issue of property rights for example social liberals are right-wing, but I'm not sure it's as helpful as it at first seems to portray social liberalism in such a reductivist way, i.e. as being simply just another solidly right-wing movement (esp. historically this would be harder to argue, considering even the very origin of the term "left" refers to liberal revolutionary sympathizers in the French National Assembly who sat to the left of the assembly's chair as compared to the royalists who sat to the right).
That said, I completely understand the reason this has become such a strongly harped-upon piece of dogma for leftists, because liberalism is so total and dominating particularly in the US, it's hard for people in the US to understand that social liberalism has any right-wing aspect, and we've come to the point now where progressives and social liberals in the US will even call themselves socialists, sort of owning the lies that the Right in America tells about them.
Also, I think on a psychological level, a lot of leftists at least in the US start as social liberals, and that process of radicalizing to the left is accompanied by feeling betrayed by liberalism, since it might be said that leftist politics is more "pure" and in accord with the values that social liberals hold and espouse. So for these people, there is a stronger desire to identify as a leftist and not a liberal precisely because of the discontent and betrayal they come to associate with liberalism.