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Well, that is the difference, you have described it correctly. A diamond doesn't have the width, but a 4-3-3 with the false 9 and wide inside forwards, who cut inside the space which was created by the false 9, have it. It is the difference. So the number 10 in diamond 4-4-2 is trequartista, and the number 10 in 4-3-3 with wide inside forwards is a false 9. These are the same things just called differently.The false 9 is a myth created by Pep to allow tactical variation because he was contractually obligated to play a 4-3-3 at the time. Its basically a 10 in a 4-4-2 diamond. The difference isn't the said "false 9" but the postioning of the forwards ahead of him. A number 10 in a 4-4-2 has 2 CFs in front of him whereas, a "false 9" has 2 wide forwards ahead of him. The actual role itself remains the same. Hence why everybody plays Cesc there since he's an amazing number 10.
As for Messi. His superhuman goalscoring instinct added to illusion of the "false 9" being an actual role when in reality it's a definition of how Messi plays. An amazingly clinical goalscoring number 10.
So to sum up, Messi's codename is "False 9" it isn't a position or role hence why nobody else can play it effectively.