With the definitions I'm using (i.e. the ones people should be using...and not just because I'm using them), it's the other way round. A scalar (field) is a function from an open subset of spacetime into the real numbers. A coordinate system is a function from an open subset of spacetime into \mathbb R^4, so each of its components is a scalar field. Proper time on the other hand, is a number assigned to a curve, not to a point in spacetime. So it's definitely not a scalar field. (Of course, if you choose one specific point p to be the starting point of all curves, and then supply additional information to narrow down the choice of curves to an arbitrary point q to one unique curve, then you can define a scalar field whose value at q is the proper time of the unique curve from p to q that meets the requirements).
What the "transformation law" for scalars \phi'(x')=\phi(x) really means is this: If \psi:U\rightarrow\mathbb R is a scalar field, and y and z are coordinate systems such that U is a subset of the intersection of their domains, we have
\psi(p)=\psi\circ y^{-1}(y(p))=\psi\circ z^{-1}(z(p))
for all p in U. Now define
\phi'=\psi\circ y^{-1},\quad \phi=\psi\circ z^{-1},\quad x'=y(p),\quad x=z(p)
and we get \phi'(x')=\phi(x).
