It's not true that this example is simple to solve via substitution; in fact it's impossible to solve these equations for $a$, $b$, $c$ and $d$. This is because $15$ different assignments of truth values to $a$ through $d$ lead to $w$ being false (namely all except the one where $a$ through $d$ are all true), and these $15$ assignments cannot be distinguished by the $8$ different assignments to $x$, $y$ and $z$, so necessarily there are assignments to $w$ through $z$ that correspond to more than one assignment to $a$ through $d$. It follows that there's no unique solution of these equations for $a$ through $d$.
As Ilmari explained in the thread you linked to, in $\mathbb F_2$ addition corresponds to XOR and multiplication corresponds to AND. You can express NOT and OR in terms of XOR and AND like this:
\neg a & = 1\oplus a\;,\\
a+b & = \neg((\neg a)\cdot(\neg b))\;.
The resulting equations over $\mathbb F_2$ won't be linear if variables are connected with AND, since that corresponds to multiplying them. As over any field, non-linear equations over $\mathbb F_2$ may or may not have a unique solution; in this case they don't.