Following Freud & Fowler, I would say that a proper joke has 3 components: malice, humor, & wit. Evidently, Fowler has a narrower definition of humor than is common. He says:
Motive or Aim: Discovery. Province: Human nature. Method or Means: Observation. Audience: The sympathetic.
And for wit:
Motive or aim: Throwing light. Province: Words & ideas. Method or means: Surprise. Audience: The intelligent.

Leper & whore: Malice: making light of suffering. Humor: none. Wit: pun on "tip". So also with the other leper "jokes", except that "strained" is a better pun.

Ant & elephant: Malice: disdain & rudeness. Humor: people can exaggerate their own importance, even when trying to be polite. Wit: matching clauses.

Barber: Malice: rudeness. Humor: barbers often take advantage of their captive audience. Wit: surprising reinterpretation of "how".

Coolidge: Malice: none. Humor; none. Wit: reply exemplifies its own truth.