What is the type number of a discrete time system given $H(z)$?
I couldn't find a definition of system type for discrete-time control systems, but in analogy with continuous-time systems you should count the number of poles at $z=1$, not at $z=0$, because in discrete time, $z=1$ is the DC point, just like $s=0$ in continuous time.

A discrete-time integrator (an accumulator), has a pole at $z=1$. Its transfer function is given by

$$H(z)=\frac{z}{z-1}\tag{1}$$