I know enough linguistics enthusiasts who feel that learning new languages makes you think in new ways that I believe that to be their genuine experience. Also because I personally find a slight difference in the way I think in different languages, though not as pronounced as those people.

Sure, but the same happens with living languages as well.

not having studied Latin enough

I studied Latin for five years. Sure, it is possible that if I had studied it longer it would have changed my thought patterns more, but surely there are cheaper ways of doing that. (Even the first couple months of studying linear algebra affected me more, but I don't expect that to apply to everybody so I didn't list it upthread.)