Minimal specification of Martin-Löf type theory
The purpose of the system described in the appendix of the HoTT book is to present something that corresponds to what is used by the book. The book is aiming to be educational. Therefore it would be a bad idea to do everything in a minimalist way. For example, we introduce $\mathbb{N}$ separately because it is instructional to see how inductive constructions work in a familiar case.

You are perfectly correct, to jump-start inductive types from general $W$-types you just need $0$ and $2$. You immediately get $1$ as $0 \to 0$, and you get $+$ from $2$ and $\Sigma$. Once you have that, you get all finite sums $1 + 1 + \cdots + 1$. At this point it is easy to do the usual algebraic datatypes.

If you drop $0$, so you start from $\Pi$, $\Sigma$, $1$ and $2$, then you cannot get $0$ back because every type you make will be inhabited.

Suppose you only have $\Pi$, $\Sigma$, $0$ and $1$. Then you cannot make $2$ because you can show that every construction you make gives you back either $0$ or $1$. In fact, you cannot make any interesting dependent families at all. A larger family of types which is closed under $\Pi$, $\Sigma$, $0$ and $1$, but does not contain $2$ is the $(-1)$-types (propositions).