Does a truism need to be true to be a truism?

Solution 1:

Truisms need only have apparent truth to earn their name, but occasionally the seeming certainty of this truth comes solely from the popularity of the saying itself.

This is, generally speaking, the logical fallacy of argumentum ad populum.

From the Wikipedia article:

In argumentation theory, an argumentum ad populum (Latin for "appeal to the people") is a fallacious argument that concludes a proposition to be true because many or most people believe it. In other words, the basic idea of the argument is: "If many believe so, it is so."

"Widely believed ideas, which really aren't true" is exactly the sense of "apocryphal" which, according to ODO, means: "of doubtful authenticity, although widely circulated as being true."

Lastly, I would refer to "widely believed ideas, which really are true" as "common knowledge."