The length of the shortest vector in Lattice
Remember that we are considering $q$-ary lattices, as defined by

$$ \Lambda^{\perp}_q(\mathbf{A}) = \{ \mathbf{y} \in \mathbb{Z}^m : \mathbf{A} \mathbf{y} = \mathbf{0} \mod q \}. $$

Clearly the vector $(q,\mathbf{0})$ is in this lattice, and has length $q$. This is a trivial, uninteresting solution, as it doesn't give a better understanding of the lattice.

$\min \{ q, 2^{2 \cdot \sqrt{(n\cdot \log{q} \cdot log{\delta})}} \}$ refers to the shortest vector one can find using (then) state of the art lattice reduction algorithms. If you're choosing parameters in order to make a cryptographic scheme secure, you are trying to ensure that the shortest non-trivial vectors that can be found are not very short. If trivial vectors break your scheme, it is broken anyway, and no amount of parameter tweaking will fix it.