Usage of imagined content in direct speech
Solution 1:
I don't think I can do better than to quote the Wikipedia page, "Quotation marks in English", which says:
Single or double quotation marks denote either speech or a quotation.
A quotation needs a source. Speech, though, just needs a defined speaker.
There IS an exception for paraphrases described:
Quotation marks are not used for paraphrased speech. This is because a paraphrase is not a direct quote, and in the course of any composition, it is important to document when one is using a quotation versus when one is using a paraphrased idea, which could be open to interpretation.
If Hal says: "All systems are functional", then, in paraphrased speech:
Incorrect: Hal said "everything was going extremely well".
Correct: Hal said that everything was going extremely well.
Seems to me that the rationale given there is incorrect, however. This is nothing to do with accuracy of attribution, and all about how a paraphrase is in a different voice; the pronouns mean different things inside the quotes as out.
So in your example:
Bob: I think I heard him say, "I like you".
...the quotes are appropriate even if it's a misquote, because it's still wrapping speech and assigning the quoted pronoun "I" to the speaker, "him", and not to Bob. If Bob heard right, 'he' likes 'you'. Whereas,
Bob: I think I heard him say, I like you.
... has three unquoted "I"s, all meaning Bob. If Bob heard right, 'he' thinks Bob likes 'you'.
To be a paraphrase in the sense of that exclusion, Bob would need to fiddle the pronouns:
Bob: I think I heard him say, he likes you.
So a more general rule might be: quote marks are required for any phrase where "I" would mean something different to the surrounding text. In which case, both of these are OK:
Bob: I said, "I like you".
Bob: I said, I like you.
Solution 2:
I think you may have conflated two uses of quotation marks. I use the first is to indicate that the quoted words are not mine, and the second is to report direct discourse. The overlap is a proper set of each. That is, there are other people's words that I wish to use but that don't appear in conversation, and there is also my own directly reported speech.
The Chicago Manual of Style dictates that in-line quotations of poetry are quoted with a "/" to separate them. The example from section 10.10 is from Andrew Marvell's poem "On Paradise Lost" and which example I adapt here:
As the poetic expression about Milton has it, "Thou has not missed one thought that could be fit,/And all that was improper dost omit." But I disagree and would change this to "Thy poem was the longest piece of shit/It seemed there was nothing thou would omit."
Two things -- 1) in the first stanza, I'm not quoting Marvel's speech; I'm reproducing his work, and 2) the second stanza makes it clear that I, at least, don't think it's true.
One more example. Do you remember when Hamlet said, "To be or not to be. Whatever."? Now we've got a misquote from a fictional character. Still in the form of a report of direct discourse. Still quoted
Solution 3:
The following would be the way to use quote marks (U.S. conventions) in the context you described.
Mary: "I cleaned him out." Joe: "Wait, do you mean, 'I won all of his money, so now he has none?'"
Even if Joe is only imagining what Mary said, he's imagining it as a direct quote. If Joe were imagining it as an indirect quote then it wouldn't have the quote marks:
Joe: "Wait do you mean that you won all of his money so now he has none?"
Some might differ on where that end question mark should go: after the single quotation mark? Typographically, this looks best to me, but logically I suppose it should go after the single quote mark.
Solution 4:
As I've come to understand quotations are used to indicate the source of the words differs from the surrounding context. This may be to site a source of information, or is often used to indicate direct discourse.
Direct discourse refers to the quoted words of a character given by the narrator.
In this case, the source of the words is the speech of a character in the narrators world. It doesn't make a difference if the world that the narrator is describing is fictional or not fictional, the punctuation and grammar are the same.
Now the examples given in the question may be explained as follows:
Mary: I cleaned him out.
Joe: Wait, do you mean, "I won all of his money, so now he has none"?
Here Marry narrates her actions. Then Joe asks whether a different phrasing would have narrated her actions just as well. In this instance the phrasing is being uttered by a hypothetical Mary, in the hypothetical world that Joe is narrating where Mary used a different phrasing. Thus Joe is really using Direct Discourse to indicate the speech of a hypothetical Mary.
Similarly, in the second example Bob is explaining that in the hypothetical world that corresponds to Bob's perception of the world, the unnamed male actually said the words, "I like you." Thus, since the words are directly quoting a source (regardless of whether it is a hypothetical source or not) they should be quoted.