My question is about c.
As per the definition, a difference operator $L_hU_m:=-a_mU_{m-1}+b_mU_m-c_mU_{m+1}$ is positive type if $a_m\geq0$, $c_m\geq0,$ $b_m\geq a_m+c_m$, and $b_m>0$.
Application of central difference for both the first and second order derivatives, I get $u''\approx h^{-2}(U_{m+1}-2U_m+U_{m-1})$ and $u'\approx 0.5h^{-1}(U_{m+1}-U_{m-1})$. So calculating the difference operator, I get $$a_m=h^{-2}+10h^{-1}\qquad c_m=h^{-2}-10h^{-1}\qquad b_m=2h^{-2}$$ So from this, I can say that the operator is of positive type. But how does it relate to the $M$? What would be its values for the operator to be positive type?
According to the definition given here, citing this original work, you have to make sure that $a_m \geq 0$, $b_m\geq 0$ and $c_m \geq 0$. Looking at your coefficents, it is clear that $a_m, b_m$ always fulfill this non-negativity condition, while for $c_m$ you need to take a closer look: \begin{align}c_m = \frac{1}{h^2} - \frac{10}{h} \overset{!}{\geq} & 0 \\ \Leftrightarrow 1& \geq 10 h \\ \Leftrightarrow h &\leq 0.1 \end{align} Thus, for $h = \frac{1 - 0}{M - 1}, M \geq 2 $ you need $M \geq 11$ to have a positive type scheme.
I am not sure why the definitions of postive-type schemes do not agree in this case.