Does "higher symmetry" mean any symmetry group of higher order?
"Higher symmetry" per se does not mean anything, unless you define it to mean something.

Depending on the context, it usually means one of the following things:


Any symmetry that is a supergroup of the group in question. In this case $C_{4v}$ is higher than $C_4$, while $C_6$ is incomparable to both. Bear with it. Partially ordered sets are quite a thing in math.
(rarely used) Any symmetry group having more elements than the group in question, so that $C_{4v}$ is higher than $C_6$, which in turn is higher than $C_4$.
(apparently, the meaning used by Cotton) Any symmetry group which is not a cyclic rotation group. In this case $C_{4v}$ is higher, while $C_4$ and $C_6$ are lower.


See, there is an intuition, after all. Worse yet, there are quite a few different kinds of intuition.

So it goes.