So, how can one say that in Esperanto? Does “Kion vi signifas?” convey the same meaning or this won’t work?
Or am I being too picky and there is no difference?
More precise would be "Kion vi celas diri?" but you'd drop "diri" in colloquial language.
Yes, Kion vi volas diri means both, though you can be more specific if you like. You can't say Kion vi signifas in this sense.
That's the tl;dr, now for the longer answer.
But nothing to correspond to 2. So in any case "Kion vi signifas" is not possible in the meaning you require.
Kion vi signifas? is wrong. Signifi means "to have a certain meaning". Words, signs, symbols, etc. have meanings, but people don't: they have intentions.
I wouldn't say that "Kion vi volas diri?" expresses that you didn't understand, but it perhaps does have the implication that the person didn't say what they meant, and invites them to change the meaning, rather than express it in a different way. I think "Kion vi volis diri?" says what you want.
Note that in English, his meaning means "intended significance" and its meaning usually means "literal significance". The former is translated celo, "aim" (or, to avoid ambiguity, you might make that dircelo) and the latter is signifo (or pedantically, denotacio).
"What were you just trying to say?"
Kion vi ĵus celis diri?
"What is the meaning of what you just said?"
Kio estas la signifo de (tio) (kion vi ĵus diris)?
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