I think it is just a wording issue about what a mass is... Don't worry. I will try to clarify.

The mass is a real and well measured quantity for any particle. It is thus constant. Period. This is what is called the rest mass in your comment.

Now, one can do the following exercise. First, we start from the energy of the particle (which contains the mass, the kinetic energy and the potential energy) and by having the E = mc2 relation in mind, we can calculate a 'mass' that is given by the energy over the speed of flight squared. This has nothing to do with the (rest) mass of the particle, but it often called the relativistic mass. This wording is very misleading (I even would say obsolete) as this is roughly speaking the energy of the particle.

And now you have a third kind of mass that is called the inertial mass. It can be derived by applying a force on the particle and measuring the acceleration, having in mind Newton's law (F = ma). However, it has to be equal to the gravitational mass, or just the mass of the object. This stems from the equivalence principle. Up to now, physicists are trying to find deviations here. So far, data shown the the gravitational mass and the inertial mass are equal at the 10^-13 level (we expect an improvement to 10^-15 by the end of this year thanks to the Microscope experiment).

I hope that this clarifies :)