Great puzzle. My approach to it was this:
1. The 3 statements are MUTUALLY EXCLUSIVE (one true implies others all false).
2. They are also EXHAUSTIVE (one of them must be true, or in other words they can't all be false).
3. There are only three of them.
4. Therefore assume in turn that each is true and the others false, and seek in each case a contradiction to eliminate the possibility of the adopted premise (reductio ad absurdum).
Most commentators seem to have followed this approach. But I also liked the post which viewed the sculpture from a different angle.
Mutual exclusivity and exhaustiveness don't just apply to the statements to be logically analysed. They also apply to the value of, say, Z (it must be either red, blue or white; and it can't be more than one of these) and they apply to the colours (red, say, must be one of the counters, X, Y or Z; and it can't be the colour of more than one of them).
Similar duality in the lines of attack apply in puzzles like Sudoku. A move can be either position-led (what value goes into this cell?) or value-led (where can 7 be placed in this column or row or box)? In my experience, once you get up to speed, most Sudoku solutions are about 80% value-led and 20% position-led, though it varies a bit from puzzle to puzzle. I suspect it also varies from solver to solver, as people develop their own preferred attack styles and special personal 'hooks'.
I often finish off Sudoku puzzles abandoned by others left on tube or pub newspapers, and try to work out their reasoning errors. A lot of less adept Sudoku solvers adopt poor notation practices (or none at all) and confuse statements about values with statements about positions. They are actually quite different, and a statement about one tells us sweet jack about the other.