Unfortunately this attempted explanation makes a common mistake. Matter is not converted to energy.
What the equation says is that mass implies the presence of a type of energy, called rest energy. Both mass and energy are properties of physical systems. Neither is "stuff". You can't have a jug of energy, nor can you have a jug of mass. Any system comprised of matter that gains or loses energy will also gain or lose mass, according to the formula. So yes, your battery will gain or lose mass when charged and discharged. Note that both mass and energy are gained and lost together. They are not converted from one to the other. They go hand in hand.
The formula is an incomplete, short, form of the full formula, which is E² = (mc²)² + p²c², where p is momentum (as measured from some frame of reference). So energy can be present due to momentum as well as due to rest mass. This accounts for the energy in radiation, which has no mass.
If you have a nuclear reaction in which rest mass is lost, it just means some of the rest energy has been converted to energy in other forms, including energy in radiation and kinetic energy of particles. Just as when you discharge your battery, it too loses mass due to the release of electrical energy.