Which of the two definitions of mass is the correct? I have searched the internet for the definition of mass when looking up about the difference mass vs inertia. There are two definitions of mass the I see: "Mass is the quantity of matter in an object," and "It is the measure of an object's resistance to acceleration." Which of these definitions is the correct? Or are they both correct and mean the same thing?
 A: In Newtonian physics, there are two definitions of mass, united only by assumption (The equivalence principle):
1. Gravitational Mass:
This is the "gravitational charge", namely the proportionality factor between the gravitational field and the resulting force experienced by a body.
2. Inertial Mass:
This is the proportionality factor between the force experienced by a body to its acceleration.
The equivalence principle states they must be the same (or proportional, if one wants to describe the two with different units).
The equivalence principle is assumed (put by hand) in Newtonian physics. If it isn't, then acceleration in a gravitational field would not be universal and there would be no free-fall (well, it would not be "free" anymore).
On the other hand, it is predicted in General Relativity, where it results from the physical equivalence of all coordinate systems (i.e., covariance under general coordinate transformations, a.k.a. spacetime diffeomorphism).
That is why experiments probing general relativity are aimed at measuring the ratio of inertial-to-gravitational mass (see, for example, the Eötvös experiment).
A: The answer by Lupus is fine. This is a clarification on terms used.
One has to keep in mind that physics is about modeling observations and measurements with mathematics. Mathematics is rigorous, depends on axioms which are assumed at the start, and are not provable. Axioms can turn into theorems but this means a theorem will become an axiom in the closed mathematical system.
Physics does not need all the functions of the mathematical system it uses, in mechanics differential equations. Physical models impose extra "axioms" called postulate, or laws , or principles, which cannot be disposed of. They are necessary in  order to pick the subset of mathematical functions that fit observations and are predictive of new set ups.
In classical mechanics the physics "axioms" are Newton's laws of motion and in classical gravitational theory, Newton's laws of gravity..
Notice that both sets of laws involve something called "mass". So mass is a quantity that is given so that the final model will fit the data.
Mass from ancient times was identified with matter, and assumed to be conserved.  It is an assumption of classical mechanics that objects have a unique mass measured in scales, and that mass is a conserved quantity. In the gravitational law it is assumed that the masses used are the ones assumed in the classical mechanics laws.
