Does $\mathrm{span}\{u, u-v\}$ contain the vector $v$? Am I thinking about this correctly?  
\begin{aligned}
\mathrm{span}\{u,u-v\} &=  c_1u+c_2(u-v)\\ &= c_1u + c_2u - c_2v \\ &= c_1u + c_2u + c_3v
\end{aligned}
where $c_3=-c_2$. So we have  $$\mathrm{span}\{u,u-v\}=(c_1+c_2)u + c_3v$$ which is just $\mathrm{span}\{u,v\}$ isn't it?  Am I making a huge conceptual mistake here?
Thanks in advance.

Your reasoning is correct.
However you notation is sloppy, we have
$$
\text{span}(v,v-u) = \{ c_1 v + c_2 (v-u) |\text{ for arbitrary } c_1, c_2 \}
$$
and we do not have
$$
\text{span}(v,v-u) =  c_1 v + c_2 (v-u) 
$$
the LHS is a vector space whereas the RHS is a vector.
