Of the lines provided on your list, the one that seems almost certainly to introduce an anecdote is option "b." However, because the information provided is limited, option "a" might be used that way.
As you note, an anecdote is a small story. Option "b" begins with a description of a man and a child. It seems likely that more information about them will follow.
The definition of anecdote you offer says it is related to the topic of the larger text. The relevance, however, may not be readily apparent. Often an author will introduce an apparently unrelated story, perhaps to draw a sharp contrast with the main topic, and follow the anecdote with an explanation of its relevance. This is actually common in persuasive rhetoric when the author or speaker thinks the reader has a different point of view and wants to offer an apparently sympathetic perspective. It is also very common in humor. A joke is often an anecdote.
Option "a" is possible, though far less likely than "b," because it seems to anticipate a "however" in the next sentence. The author could follow his statement with "however, I once knew a woman who was so superstitious that..."
Option "c" is very unlikely to be the type of rhetorical device that will lead in to a story. Further emphatic rhetoric, perhaps used in oratory, is likely to follow.
If you look at the choices that you have given us, only one of them is in any way an anecdote. That is B. The others are not anecdotes at all.
As you say in your question, an anecdote is like a little story that you tell to illustrate a point that you are trying to make. Only B is a story. In A and C, Paine is simply stating his opinion. Stating your opinion is not an anedote. But in B, Paine is telling a story. It happens that he is telling a story about a Tory who lived in Amboy, meaning to have the story show us the unpleasant attitudes that such people have.
Therefore, the answer has to be B.