Yes. The meaning of b'farhesia.
With all due respect to the research you put into this very sophisticated answer. This is not the Ben Shapiro show where if it's in the name it's good enough.
Halacha doesn't work that way. As an example (not sure what your standards are) according to your theory there should be no such thing as cholov stam (a term coined in the 20th century) There should be cholov yisroel (as explained in shu"a) and non kosher milk. The name clearly says it.
Coming back to the halacha we are discussing, we need to go to the source of considering someone that is mechalel shabbos b'farhesia a goy/mumar lehachis. If you say (as most poskim) it's from the possuk v'nikdasty b'soch bnei yisroel then it would have to be in front of 10 men in a way of them standing 'together' for a minyan.
Regardless there are so many variables in such a situation for instance the m"b writes that even if he does it in front of 10 people but would be embarrassed to do it in front of a great person it is still considered in private and he would not be considered a goy/mumar lehachis.
Pls don't go and write people off of yidishkeit just on a definition of a word.