What exactly is “electrical energy”? If the two fundamental types of energy are kinetic and potential energy, is electrical energy simply the kinetic energy of charge carriers?
Also, is the statement “A cell converts chemical potential energy to kinetic energy of the charge carriers.” correct?
For example, when charge carriers pass through a light bulb, some of their energy is converted to, say, light energy. What is the nature of the energy that they lose? If it is kinetic energy, then would the charge carriers actually slow down?
 A: 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.
A: On a very broad level this energy is intrinsic to the electron, and is its own category. Each electron has a certain amount of it and when it goes through something like an LED it looses some of it.

I would not call potential energy a fundamental type of energy. I would say potential kinetic, or potential chemical, or potential electric.
