What exactly is “electrical energy”?

Electrical energy generally means potential energy.

To take your example of a light bulb (we'll assume a DC source for convenience) the electrons leaving the -ve terminal on the battery and entering the wire/bulb have a higher potential energy than the electrons leaving the wire/bulb and reentering the battery at the +ve terminal. As the electrons flow through the filament of the bulb their potential energy is converted to kinetic energy, i.e. the electrons accelerate. The electrons scatter off the atoms in the filament and transfer their kinetic energy to lattice vibrations of the filament, i.e. heat, so the filament heats up. The filament then emits photons by black body radiation. So potential energy $\rightarrow$ electron kinetic energy $\rightarrow$ lattice kinetic energy $\rightarrow$ photons.

But I would be cautious about trying to categorise energy. The word energy is used in a large number of different circumstances to mean different things, and I'm not sure that trying to pin a precise definition to it is very useful.