In natural units (mechanically) ##E = \sqrt{p^2c^2 + m^2 c^4}##.
In the geometrized system, distance and time are not dimensionless, but they both are set to ##1##, which makes speed dimensionless. That makes the speed of light dimensionless, too, so you can set it to ##1##, too. When you lose dimension, and by fiat set ##c=G=1##, you can as a consequence , instead of saying ##E=mc^2##, say ##E=m##.
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One problem with that is that the resulting 'equation', without units, tells us nothing about
how much energy is equal to
how much mass.
It's not problematic that both energy and mass are defined along ##[L]## (length); however, if we want to quantify, we need to define how many meters of energy is equal to how many meters of mass, not just by an equal number, but also in terms of some other fundamental factual quantity, such that ##E=m## for only 1 unique value on that reference scale.
Even though ##E## and ##m## both use the length dimension ##[L]## and are both set equal to ##1## in the geometrized unit system, the multiplication factor (for conversion between geometrized and SI) for ##E## is ##G c^{-4}##, while for ##m##, it's ##Gc^{-2}##.
When we say that ##c = 1, \hbar = 1, G = 1##, we should remember that these are not really equalities. In each of them, only the RHS is
properly dimensionless; the LHS is not dimensionless. The fact that we can simplify some equations by using these pseudo-equalities to eliminate some terms does not properly make the SI versions of the units meaningless.
The loss of dimensional information that allows setting ##c=1## means, among other absurdities, that ##c=c^2##, which entails the nonsensical notion that velocity is the same as acceleration.
I think what is really meant by ##c=1## is not actual equality, because in that 'equation', the LHS still factually has dimension, and only the RHS is dimensionless, so it could instead, I think more properly, be written as ##c \mapsto 1## or ##c \to 1## (i.e. ##c## 'maps to', or more precisely, 'is substituted for by' ##1##).
When converting back to SI units, in order to be able to present meaningful numerical results, we have to 'unforget' the ##c^2## coeffficient, and so return from our temporary sojourn in which we used ##E=m##, back to the more familiar and realistic ##E=m{c^2}##.