So in the end, this stuff gets largely uncorrected.
I have long found the fact that 'trying to be smart' is an insult in our society, to be a source of both bewilderment and depression. I remember being utterly confused when even the teachers would use it at school. I'd ask them why we were at school, if we were not trying to be smart? Which would then get me into trouble, presumably for trying to be smart.
After a while I got bitter about this and my response turned into; 'Yes I am. Tell me, how is trying to be thick working out for you?', which still got me into trouble, but with more entertaining results.
Note however that in both cases they (and generally all humans, to some extent) care about their relative status; in most circumstances (especially when in a group), correcting someone will be seen as an attempt to diminish their social status.
It's unfortunate that this is the case, and a lot of people who honestly value being correct have to struggle with getting the handle on this whole status mechanism. I'm not even blaming most people that much here; I do my best to always accept corrections well, and I still at times feel bad or get angry when someone points out that I'm talking nonsense.
In the moment, I can be very combatative and do not tend to fold easy, but if someone demonstrates to me conclusively that I am talking utter bollocks, I appreciate that I have learned something and I try to remember to thank them.
Am from the UK and I have heard that story for years from a variety of people who do seem to completely believe it, as it is usually brought up to give an example that shows Americans being stupid, and the motive of 'aren't we better' would not be there if they thought that the story was false. I didn't know the truth of it but thought it was a funny story, so had gone to look it up a while back.
Is also usually brought up by the same people who will crow about Americans not understanding the concept of irony or sarcarsm, again to demonstrate the supposed superiority of UK culture. I have taken to pointing out that saying that tells me nothing about Americans, but does demonstrate that the person I am talking to presumably never reads any books.