In this thread there have been a number of odd statements. I can't figure
out which are facetious and which are merely befuddled.
In current professional usage, "mass" means "rest mass". Indeed, saying
"rest mass" is a pleonasm, so I'll stop saying it. The "m" that appears in
equations like E = m c^2 is this mass.
Therefore the aforementioned famous formula does not describe the total
energy. Far from it. The total energy includes contributions from:
-- the mass (as described above),
-- the kinetic energy (which is not included in the mass),
-- the gravitational potential energy,
-- the electrostatic potential energy, and
-- many other things.
To say it another way: Mass is not "equivalent to" energy or "identical
to" energy. It is just one contribution to the energy.
If you want to deal with particles not at rest, a useful formula is
E^2 = p^2 c^2 + m^2 c^4
and if the students can only remember one relativistic formula, IMHO this
is the one you want them to remember. Draw a graph of E versus p.
This is elegant and has some nice properties:
a) it reduces nicely to E=mc^2 in the at-rest case;
b) it reduces nicely to E=pc in the massless case; and
c) to lowest order it says KE = p^2 / 2m for a slowly-moving massive
particle.
There is very little professional use of the quantity "gamma m".