What is the meaning of $O(m+n)$?
You are right. Notice that the term $O(n+m)$ slightly abuses the classical big-O Notation, which is defined for functions in one variable. However there is a natural extension for multiple variables. 

Simply speaking, since
$$ \frac{1}{2}(m+n) \le \max\{m,n\} \le m+n \le 2 \max\{m,n\},$$
you can deduce that $O(n+m)$ and $O(\max\{m,n\})$ are equivalent asymptotic upper bounds.

On the other hand  $O(n+m)$ is different from $O(\min\{n,m\})$, since if you set $n=2^m$, you get $$O(2^m+m)=O(2^m) \supsetneq O(m)=O(\min\{2^m,m\}).$$