Derivation of the transfer function form with the system type number included
The type number is the number of pure integrators in the forward path, \$\small G(s)\$, of a closed loop system. This is equivalent to saying the number of 'free' \$ s\$ terms in the denominator of \$\small G(s)\$ . Alternatively, the number of poles of \$\small G(s)\$ at the origin.

It can easily be shown that the type number defines the steady state response of the closed loop system to various deterministic input signals. Thus a type 1 system will have zero SS error to a step input, a type 2 will have zero SS error to a ramp input; a type 3 to a parabola etc.

Hence if the CLTF is \$ \frac{G(s)}{1+G(s)H(s)}\$, then the type number is the number of denominator free \$s\$ terms in \$\small G(s)\$. Note, type number is not determined by the open loop TF, \$\small G(s)H(s)\$, unless the system is unity feedback.