From Wikipedia, there are 5 different kinds of lattice in 2 dimension:
But I am wondering how the third type (the centered rectangular) is different from the first kind (the oblique lattice). The unit cell is the same in both, except that we change from $|\alpha_1| > |\alpha_2|$ to $|\alpha_2| > |\alpha_1|$. I find no one asking this problem online so I posted here.
The lattices differ from each other in the amount of symmetry they have. You are correct that lattice $\mathbf{3}$ is basically the same as $\mathbf{1}$, but then all the lattices are basically the same as lattice $\mathbf{1}$ just with extra symmetry. For example the square lattice is derived from $\mathbf{1}$ by requiring that $a_1 = a_2$ and $\phi = \frac{\pi}{2}$.
In the case of lattice $\mathbf{3}$ you derive it from $\mathbf{1}$ by requiring that $a_1\cos\phi = \tfrac{1}{2}a_2$ which gives you the rectangular symmetry. This extra symmetry is the difference between the two lattices.
