What does that phrase mean to you? Is Wikipedia correct in saying it "indicates doubt about someone's sincerity?"

I'd say that's right. One can refute one's sincerity by overdoing something, as is the case in Hamlet. Gertrude, however, doesn't see herself in it, IIRC.

I always have taken it to mean that the person who is protesting is protesting so much, that maybe they are guilty of what they are blaming the other person of. For example, Donald accuses everyone of "fake news," knowing that he is the liar. Or some such...

I'm reminded of this: “If you don't know how to pronounce a word, say it loud! Why compound ignorance with inaudibility?” William Strunk

I don’t really see any great difference between those two interpretations.

I agree with Bob. Kalleh, your interpretation is modern, informed by psychology: you recognize the character may be "sincere" in a technical sense, but her words display subconscious insincerity.

Raises the question of whether it is logically possible to be sincere about something you don’t actually believe. You can certainly appear sincere but that’s not the same thing at all.

Bob, give me an example. I can be sincere about other religions (other than Judaism), but I don't believe in them. I am trying to understand your perspective.

Let me just rephrase. Is it possible to sincerely claim something that you don’t actually believe?

Can't one can play devil's advocate with sincerity? Don't trial lawyers do it all the time?

I did say that appearing to be sincere isn’t the same thing as being sincere.

I suppose you are right about that, Bob. I couldn't, for example, convince anyone that I don't believe in abortion.

As for trial lawyers, they are standing up for their legal principles: Even if they believe their client is guilty, they must fight for them.