The difference relates with the excited modes versus the possible modes.
2 "tumbling" modes can only be the rotations. 3 translations and 2 rotations for a diatomic gas molecule means that this author has neglected the vibration mode. This is correct for N2, H2, O2... at room temperature, because the vibration energy levels for these molecules are too separated, so the ambient heat doesn't excite them, and the molecule stays in its ground state. Gaseous Br2, as opposed, vibrates more easily, so its mode is largely excited and stores heat too.
At heat, stiff modes gets excites too. Cl2 is an intermediate case, with its vibration little excited at room temperature but much at heat.
Notice that each well-excited vibration mode stores R per mole, while translations and rotations store only R/2.