Confusing notation in concatenation of error correcting codes
You get it right.

The notation $$(C_{out}\circ C_{in})(m)$$
means: "take m, encode it with $C_{out}$ and then encode each symbol with $C_{in}$."
From the point of view of the channel, $C_{in}$ is "closer", hence the inner one; see the figure in the wiki page you refer to.

This is indeed what the wiki page says.  

The confusion is probably because you think of composition of functions, where the common notation $(g\circ f)(x)$ means $g(f(x))$. But here we don't have function composition but instead code concatenation, so the $\circ$ operation has a different meaning - it means
$$ C_{in}(c_1), \ldots, C_{in}(c_n)$$
with $(c_1, \ldots, c_n) =  C_{out}(m)$.    

Just to complete the picture, concatenation codes where developed by David Forney in 1965, where he named the inner/outer codes that way (again, having the channel  as the center of this point-of-view). See his PHD work and in particular Figure 2 that illustrates this concept (same as in wikipedia).