Dear Andrew, your prescription for $\mathcal K'(U)$ does NOT yield a presheaf: you cannot restrict an element of that ring to a smaller open subset $V \subset U$ because non zerodivisors do not restrict to non zerodivisors. Don't feel bad about this error, you are in good company: Grothendieck, Kleiman and Hartshorne (among others) made the same mistake.Kleiman saw the light and wrote an article aptly named
Misconceptions about $K_X \quad $ (L'Enseignement Mathématique, 25(1979), 203-206)
where he gives a correct definition. He addresses your question by constructing a beautifully geometric (but sophisticated) example of an affine scheme X=Spec(A) where $\Gamma(X,\mathcal K) $ (with the correct definition of $\mathcal K$ !) is strictly bigger than Frac(A).
A gift from L'Enseignement Mathématique Our generous Swiss friends allow us to freely download all issues of their journal from 1899 to 2004 [click on Tome 1(1899)-50(2004), in the green column on the left]:
Kleiman's article in particular is here [take the line correponding to page 203 and click on the white PDF logo on the left]left]: