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Your confusion should be removed away when you consider the fact that the following signals are the same : $$u[n] - u[n-3] ~ = ~ \delta[n] + \delta[n-1] + \delta[n-2]$$ or generalizing for any integer $M$: $$u[n] - u[n-M] = \delta[n] + \delta[n-1] +...+ \delta[n-M+1] = \sum_{k=0}^{M-1} \delta[n-k]$$ or even further for $K < M$  u[n-K] - u[n-M] = ...
Taking Equation $2.74$ from Example $3$ and setting $M_1=0$ gives: $h[n] = \frac{1}{M_2+1} \sum_{k=0}^{M_2} \delta[n-k]$ Now lets take a closer look at this: $h[n]$ is non-zero only for certain values of $n$. Its a good a starting point as any, so lets look at $n=0$. We have: \$ h[0] = \frac{1}{M_2+1} \bigg( \delta[0] + \delta[-1] + ... + \delta[-M_2] \...