Question: Are inertial mass and gravitational mass identical, or just indistinguishable? If they are identical, why do different definitions of mass exist?
Newtonian mechanics relates mass to force and acceleration with the equation F = ma. This is inertial mass because the more of it a body has, the greater the force required to achieve unit acceleration.
According to the Wikipedia article Mass, there are two other kinds of mass, active gravitational mass (more of this makes a body attract other bodies more) and passive gravitational mass (more of this makes a body more attracted to other bodies). I don't really understand the distinction, so I'll just consider them together as gravitational mass and define them by Newton's law of universal gravitation, Equation 1:
Combining the two equations yields Equation 2:
Equating m with m1 yields Equation 3:
Factoring out m1 gives Equation 4:
Rearraning then leads to Equation 5:
In words, Equation 5 says that the mass, m2, of a body is equal to the acceleration the body exerts on a test mass (the mass of which, m1, does not affect m2) multiplied by the square of the separation between the body and the test mass, and divided by the gravitational constant, G.