The Point IS...
that you can treat the energy of the object like the mass is increasing, or you can accept that the energy of the object increases non-linearly with increasing velocity. This way, you don't have to resort to a nonphysical interpretation just because the math happens to work in the basic example of a point particle. Here's a hint, it don't work so nice for more complicated systems.
Don't just regurgatate a page full of crap I can pull from any General Physics book at me when I'm trying to explain a subtle point so we can ALL learn something here.