I'm so confused due to how many different equations for energy are there. For example:
E = m$c^2$,
E$^2$ = (m$c^2$)$^2$ + (pc)$^2$,
E = mγc$^2$ - mc$^2$
What does each equation represent? What is the relativistic kinetic energy? Why is m and $m_0$ sometimes used interchangeably?
The equation for the total energy of a relativistic object is:
$$ E^2 = p^2c^2 + m^2c^4 \tag{1} $$
If we consider only massive particles, so this excludes photons, the relativistic momentum is $p=\gamma mv$ we can rewrite equation (1) as:
$$ E = \sqrt{\gamma^2m^2v^2c^2 + m^2c^4} $$
Then take out a factor of $mc^2$ to get:
$$\begin{align} E &= mc^2\sqrt{\gamma^2\frac{v^2}{c^2} + 1} \\ &= mc^2\sqrt{\frac{v^2/c^2}{1-v^2/c^2} + 1} \\ &= mc^2\sqrt{\frac{v^2/c^2 + 1-v^2/c^2}{1-v^2/c^2}} \\ &= mc^2\sqrt{\frac{1}{1-v^2/c^2}} \end{align}$$
Which is just:
$$ E = \gamma mc^2 \tag{2} $$
So for massive particles the equations (1) and (2) are equivalent. However for massless particles equation (2) is no use while equation (1) still applies. That's why most physicists will use equation (1) though you will still see equation (2) from time to time.
Finally, if the total energy is $E$ you can subtract off the rest energy $mc^2$ to give the equation for kinetic energy you have as your third equation:
$$ KE = \gamma mc^2 -mc^2 \tag{3} $$
Your first equation, the famous $E=mc^2$, is the total energy of the object in its rest frame i.e. the frame in which its velocity is zero.
