I thought I understood the idea behind a Schur number but now wonder.
$S(2) = 5$ which, I think, means that for every partition of $1...5$ into two non-intersecting, non-empty sets at least one would have the triple $x + y = z$.
So I can see that, eg $ \{1, 2, 3\}\{4, 5\}$ works, as does $\{1, 4, 5\}\{2, 3\}$ and $\{2, 3, 5\}\{1, 4\}$, but what about, eg, $\{1, 2\}\{3, 4, 5\}$ and $\{1, 5\}\{2, 3, 4\}$?
It appears I've misunderstood the definition in some way: could someone explain in what way?
$x,y,z$ don't have to be distinct for the theorem's purposes, so in your last two colourings you have monochromatic $1+1=2$ and $2+2=4$ respectively.