As far as I understand from what I'm studying, only when you have more than one kind of atom in the lattice structure, will the phonon have an optical branch, or else it should not have an optical branch, because as was written in the text, "two atoms per unit cell and more". Am I interpreting it correctly or am I being mislead? If I am correct, then why does Silicon have an optical phonon branch?
Just to be clear, the two or more atoms do not have to be of different type. Optical phonons are related to the relative vibrations of atoms within the unit cell, while acoustic phonons describe the relative vibration of different unit cells. Optical phonons arise whenever the unit cell has at least one such degree of freedom, meaning at least two atoms in the unit cell. This includes silicon.
Crystal lattices are classified in a way that is not necessarily the most natural one. A first classification is by their lattice class, which is determined by its associated unit cell, which is not necessarily the same as a fundamental domain for the lattice. There can be different Bravais lattices in each lattice class, determined by possible additional vertices (locations) within each unit cell. This still only depends on the vertex locations, not the types. In the case of silicon, the crystal lattice is a face centered cubic lattice, whose unit cell contains two atoms.