What is a Joke?

A joke is an attempt to be humorous, but it can also be--as you point out-- inadvertent, accidental, or unintended.

At the heart of a joke, of course, is humor (BE, humour). Humor could be defined as a "sudden happiness increment caused in part by a surprise element, characterized by incongruity." (I take the blame for this definition, which I've cobbled from memories of an ancient study I did on irony.)

So, if the "joke" is unintended, does it still count as a joke? I say no. While I am at a loss for a single word to use (if there is one) to label the unintended happiness increment, in order for there to be a joke, I feel, the implication of the joke teller and the inference of the joke hearer need to be congruous. In other words, the teller and hearer need to be on the same wavelength, as it were.

If they are not, then what you have on your hands is a dud or "failed to launch," to change the metaphor. In comic parlance, the joke "died" or failed to "kill." Put differently, the audience isn't picking up on what the joke teller is laying down!

On the other hand, if a speaker says something which turns out to be unintentionally funny (i.e., there is no deliberate implication on their part), the incident could turn into a joke in at least two ways, but that would require a bit of evolution. By that I mean the hearer could coopt the perceived element of humor, add or subtract an element (or elements) to it, and then tell it as a joke. There is, of course, no guarantee it will create a happiness increment!

Finally, the unintended humor could turn into an anecdote, which is not strictly speaking a joke, but rather a story or narrative with a humorous element. Either way, the re-worked joke and the anecdote can fail to arouse a happiness increment!

All language works this way - this is what has been called Prototype Theory. The core prototypical sense of a word remains the same, but we apply it to more and less prototypical things. Just because we have step-father, grand-father, spiritual father, the father of modern science, the father of Star Wars, all with different connotations, doesn't mean that the meaning of the lexical item father changes, it still means "a male parent". You don't have to account for these extensions of the meaning, because we recognise that they are extensions, not the core prototypical meanings. And the fact that we often use modifiers like step- helps us know that we're not using the prototypical meaning.

So as Jacinto said in a comment, the first is a misunderstood joke and the second an unintended joke. The prototypical meaning of joke is still "something said to provoke laughter", and those modifiers explain how the meaning is changed.