Some people say that mass increases with speed while others say that the mass of an object is independent of its speed.
I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be "true" is any sense of the word.
Either mass increases or it doesn't, right?
Can't we just measure it, and find out which "interpretation" is right? E.g. by heating up some particles in a box (in some sophisticated way) and measuring their weight?

UPDATE:
Right, so I've got two identical containers, with identical amounts of water, placed on identical weighing scales, in the same gravitational field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and $g$ is constant, does this mean that the $m$ in $W=mg$ has increased?

There is no controversy or ambiguity. It is possible to define mass in two different ways, but: (1) the choice of definition doesn't change anything about predictions of the results of experiment, and (2) the definition has been standardized for about 50 years. All relativists today use invariant mass. If you encounter a treatment of relativity that discusses variation in mass with velocity, then it's not wrong in the sense of making wrong predictions, but it's 50 years out of date.
As an example, the momentum of a massive particle is given according to the invariant mass definition as
$$ p=m\gamma v,$$
where $m$ is a fixed property of the particle not depending on velocity. In a book from the Roosevelt administration, you might find, for one-dimensional motion,
$$ p=mv,$$
where $m=\gamma m_0$, and $m_0$ is the invariant quantity that we today refer to just as mass. Both equations give the same result for the momentum.
Although the definition of "mass" as invariant mass has been universal among professional relativists for many decades, the modern usage was very slow to filter its way into the survey textbooks used by high school and freshman physics courses. These books are written by people who aren't specialists in every field they write about, so often when the authors write about a topic outside their area of expertise, they parrot whatever treatment they learned when they were students.  A survey [Oas 2005] finds that from about 1970 to 2005, most "introductory and modern physics textbooks" went from using relativistic mass to using invariant mass (fig. 2). Relativistic mass is still extremely common in popularizations, however (fig. 4). Some further discussion of the history is given in [Okun 1989].
Oas doesn't specifically address the question of whether relativistic mass is commonly used anymore by texts meant for an upper-division undergraduate course in special relativity. I got interested enough in this question to try to figure out the answer. Digging around on various universities' web sites, I found that quite a few schools are still using old books. MIT is still using French (1968), and some other schools are also still using 20th-century books like Rindler or Taylor and Wheeler. Some 21st-century books that people seem to be talking about are Helliwell, Woodhouse, Hartle, Steane, and Tsamparlis. Of these, Steane, Tsamparlis, and Helliwell come out strongly against relativistic mass. (Tsamparlis appropriates the term "relativistic mass" to mean the invariant mass, and advocates abandoning the "misleading" term "rest mass.") Woodhouse sits on the fence, using the terms "rest mass" and "inertial mass" for the invariant and frame-dependent quantities, but never defining "mass." I haven't found out yet what Hartle does. But anyway from this unscientific sample, it looks like invariant mass has almost completely taken over in books written at this level.
Oas, "On the Abuse and Use of Relativistic Mass," 2005, here.
Okun, "The concept of mass," 1989, here.
