IMO both usages are correct, although yours is probably more common. 'Dubious' here means something like questionable/uncertain & it can apply either way: The distinction can be 'questionable' in terms of whether it's something to be proud of, or in terms of whether it's even true in the first place.
I had a similar discussion here with someone who thought that "dynamic programming" was the act of programming in some sort of dynamic manner. Not sure I managed to disabuse them of that notion.
A "dubious distinction" is a distinction which is undesirable.
West Virginia has the dubious distinction of being the US state with the lowest average educational achievement.