The latter is a strict subset of the former, but it is the big one if “everybody knows that Y is false”.
In the cases where asserting Y when Y is false, but it is not common knowledge that Y is false, the path to a problem likely passes through “People actually believe Y, because they have been told Y by someone who knows if Y is true.”
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The difference between the two may be relevant to the contents of the OP.
The latter is a strict subset of the former, but it is the big one if “everybody knows that Y is false”.
In the cases where asserting Y when Y is false, but it is not common knowledge that Y is false, the path to a problem likely passes through “People actually believe Y, because they have been told Y by someone who knows if Y is true.”