Regarding the final paragraph and implicit assumptions regarding the veracity of the content of the four canonical gospel accounts, I am not sure attributing it to "simple" assumption (i.e., the colloquial assumption where one just thinks something is true without examination) is reasonable given the education of both men and the body of work done over several centuries on examining epistemologically and historically the veracity of the gospels. See for example the intense work of John Henry Newman, 19th century Oxford scholar and intense defender of the historical reliability of the canonical gospel narratives.
“It is difficult to get a man to understand something, when his salary depends on his not understanding it.”
Fulton Sheen cheats by pretending to use your definition but subsuming more than deliberately lying under the liar banner. Very sneak these philosophers.
(I think we're in agreement that being mistaken about things is not incompatible with being a "great teacher", as I say in another comment here.)
It was valuable to show that Hume's approach was flawed but limited value to prove the truth per se. To apply probabilities to the events you'd have to make an awful lot of assumptions about the accuracy of the, non-eye witness, reporting. However, it was an excellent approach for such an evaluation.
As to the Fulton Sheen quote. I's a bit unfair. I mean, if you're going to be picky about it you can lose the lunatic option: someone who claims to be God is either God or a liar, by definition. But you can be a liar by deliberate act, by diminished faculty or by being mistaken. And plenty of great teachers have been mistaken about important things.
And of course, in both the Bayes and Fulton cases, there is the implicit assumption that what was recorded about Jesus had no systemic bias i.e. any errors while recording what He said and did were mistakes, not deliberate misrepresentations. Given the strong requirement to show that He fulfilled scripture, this is a highly dubious assumption.