This type of presumption drives me up the wall. How in the world do you know when people lose interest? What, everyone else has an IQ of 70? They need to hurry up to get to the end of a sentence? Give me break.
Not true. One of the links I posted to defines broken pediment as one broken at its base. Have you interviewed all architects in the US to find out what the most common usage is?
The pediment started off as a decorative motif over entrances. Since the Renaissance, it's been used to decorate entrances, windows, doors, chimneys, gable ends and a variety of other items. As a decorative motif it has its reason for use, whether supported by actual formal columns or not. The photo you posted was an attempt at that same decoration, with the horizontal cornice broken (either for looks or expediency). Hence broken pediment. Since I haven't interviewed every carpenter in the US, I can't vouch for what term they would use, but even so, given that the element in question has 2000 years (if not more) of historical usage by others, I, as a design professional (who ought to know a bit about what the pediment actually is and where it came from), would prefer to continue to use the right term, given that it's neither archaic, nor incorrect.