Comparison Procedure in Robinson's Unification Algorithm
It looks like the answer is that, to be syntactically valid, composite types must be fully parenthesized.

In particular, $a \to b \to c$ is not a syntactically valid type (per definition 2A1).  2A1.1 introduces "syntactic shorthand": when they write something like $a \to b \to c$, it should be understand that they actually mean $(a \to (b \to c))$.

The Note appears to be valid when we restrict to syntactically valid types.  Thus, your example is not actually a counterexample.