Is associativity implicated by commutativity
In your second way to solve the problem, you write the expression $a\circ b\circ c$ which is not defined.
We usually only define $a\circ b\circ c$ if we already know that $\circ$ is associative, in which case we can shorten $(a\circ b)\circ c$ and $a\circ(b\circ c)$ to simply $a\circ b\circ c$ because we already know that both of those expressions are equal. If we do not know that, then the expression $a\circ b\circ c$ is not yet defined, because it could mean either one or the other, and there is no guarantee that they are the same.
Certainly, we can define $a\circ b\circ c$ to mean $a\circ(b\circ c)$, but if you define it like that, then your claim that
$$a\circ b\circ c = b\circ a \circ c$$
becomes (using the definition) the claim that
$$a\circ(b\circ c) = b\circ(a\circ c)$$
which is not a claim that can be proven by simply claiming commutativity.
Similarly, if you define $a\circ b\circ c=(a\circ b)\circ c$, you hit a problem because then you actually can claim that $$a\circ b\circ c= (a\circ b)\circ c=(b\circ a)\circ c=b\circ a\circ c$$ but then the claim
$$b\circ a\circ c=b\circ c\circ a$$
is equivalent to
$$(b\circ a)\circ c = (b\circ c)\circ a$$
which can again not be proven usinc commutativity alone.
Your first proof is correct, other than the fact that you write (twice) “$=2c$” instead of “$+2c$”.
The other proof is wrong from the start. The expression $a\circ b\circ c$ is ambiguous if you don't assume associativity. A commutative operation doesn't have to be associative.