Does precedence matter in constructing parse trees of arithmetic expressions for postfix notation?
What you're missing is the associativity of operators. All operators you mention in your examples are binary. An expression of the form $A + B + C + D$ is interpreted by the parser as $(((A+B)+C)+D)$. When converting such a fully parenthesized expression into prefix form, we apply the rule $(\alpha + \beta) \to +\alpha\beta$. Denote this transformation by $T$. Then
$$
T[(((A+B)+C)+D)] = +T[((A+B)+C)]D = ++T[(A+B)]CD = +++ABCD.
$$
In a completely analogous way, the operator $S$ converting into postfix makes the transformation $S[(\alpha+\beta)] = \alpha\beta+$:
$$
S[(((A+B)+C)+D)] = S[((A+B)+C)]D+ = S[(A+B)]C+D+ = AB+C+D+.
$$

The operator $+$ is associative, but imagine what happens with an operator like $-$. The expression $A-B-C$ is interpreted as $((A-B)-C)$, and its prefix and postfix forms are $--ABC$ and $AB-C-$, respectively. Any other conversion would result in an arithmetically inequivalent expression. That's why associativity matters.

Most operators associate to the left, just like $+$. Exponentiation traditionally associates to the right.

On top of this, we also have operator precedence, which leads to the parsing of $A+B*C+D$ as $((A+(B*C))+D)$. If you apply the transformations $T$ and $S$ you will get the expressions given in the book.