Ok, in my opinion you must go through the following steps if you want to grasp the true meaning of the equation

#1 Before Newton people assumed that matter and weight were the same stuff, weight being a constant on the surface of the earth

#2 Newton defined momentum of a material body as p = mv
From experience the total momentum of an isolated system is a constant thus
F = dp/dt = ma ,
The mass m , being a constant, became a synonimous with matter and still it is in our daily conversation

#3 in the 19th century it was already known that also electromagnetic radiations have momentum
A beam of light can make a small wheel rotating
The light momentum is p = L/c
Where L is the energy of the radiation
This was an amazing discovery since momentum was supposed to be a mass related parameter

#4 Einstein and some others claimed :
If a material body absorbs a radiation than its momentum turns into mass given by momentum divided by speed i.e

dM = L /c^2

This is the original equation where "m" is the Newtonian inertial mass
Mass, being a variable, can not be considerd any longer as a measure of the amount of matter of a body


#5 Einstein made a step futher
In the theory of relativiy Einstein demonstrated the total energy E is the sum of two addendum : the momentum and an universal constant
If so even a massless entity such as light can exert a pressure on a surface(the spinning wheel) but on the other hand a material body has also an intrinsic non variable property which is entitled to be considered the "matter "
This parameter is the rest mass given by

E = Moc^2

#6 The confusion come frome the fact that people do not distinguish m from m0
when you claim that mass can be turned into energy you speak of m (Inertial mass)
Whe you cliam that mass and matter are the same stuff you speak of m0 ( the rest mass)

This at least is my understanding of the equation
In