Three secret numbers (Variant 1 and Variant 2)

The statements of the following two puzzles are almost identical. The only difference consists in the words printed in boldface. The two solutions, however, turn out to be totally different.

Variant 1 of the puzzle

My friends Xavier, Yvo, and Zeno are excellent mathematicians and always think strictly logically. Yesterday I told them: "I have secretly chosen three positive integers $x,y,z$ with $x+y+z=2015$. I have told $x$ to Xavier, $y$ to Yvo, and $z$ to Zeno."
Then the following conversation developed.

1. Xavier said: I know that Yvo and Zeno have different numbers.
2. Yvo said: I already knew that all our numbers are different.
3. Zeno said: Aha! Now I know all three numbers.

Determine the values of $x$, $y$, and $z$!

Variant 2 of the puzzle

My friends Xavier, Yvo, and Zeno are excellent mathematicians and always think strictly logically. Yesterday I told them: "I have secretly chosen three positive integers $x,y,z$ with $x+y+z=2015$. I have told $x$ to Xavier, $y$ to Yvo, and $z$ to Zeno."
Then the following conversation developed.

1. Xavier said: I know that Yvo and Zeno have different numbers.
2. Yvo said: Aha! Now I know that all our numbers are different.
3. Zeno said: Aha! Now I know all three numbers.

Determine the values of $x$, $y$, and $z$!

• Should we guess the numbers x, y, and z or should we determine how Zeno guessed them? – dmg Jan 28 '15 at 16:53