What does it mean to add up O-terms with different variables?
Yes, this is true.

The LHS means that for some $N$, $M$ and some constants $c_n,c_k$,

$$n\ge N,k\ge M\implies f(n,k)\le c_nn+c_kk.$$ 

The RHS means that for some $N'$, $M'$ and some constant $c$,

$$n\ge N',k\ge M'\implies f(n,k)\le c(n+k).$$

These two statements are equivalent if you take $c\leftarrow\max(c_n,c_k),N'\leftarrow N,M'\leftarrow M$ one way and $c_n=c_k\leftarrow c,N\leftarrow N',M \leftarrow M'$ the other.