I don't want to get too nit-picky, but IMHO it would be better
to emphasize that mass is best defined *without* reference to
any frame.
Let's talk about the mass of a single particle:
-- The particle exists, independent of whatever observers
(if any!) are observing it.
-- The particle exists, independent of whatever coordinate
systems (if any!) are being used.
-- The particle's momentum p is a four-dimensional vector,
and is a first-class thing unto itself, a physical and
geometrical thing, independent of whatever coordinates
(if any!) and whatever basis vectors (if any!) are being
used ... and independent of any frame.
-- The particle's mass can be defined in terms of the invariant
gorm p•p ... specfically,
m = √(-p•p) [1]
-- Equation [1] is particularly easy to apply in the rest
frame of the particle, but the equation is manifestly
frame-independent, and is very commonly evaluated in all
sorts of non-rest frames.
Just as the three-dimensional dot product is invariant with
respect to spacelike rotations, the four-dimensional dot product
is invariant with respect to four-dimensional rotations, which
includes spacelike rotations as well as boosts.
Most generally: Whenever you see a definition or an equation
that seems to be valid in some special frame only, it pays to
look around and see if there is a frame-independent way of
expressing things.
The previous sentence applies at all levels, from the most
introductory level on up, in every number of dimensions from
two on up.
As a high-school-level example of what I'm talking about, consider
the equation
ΔE = m |g| Δz
You'd think such a simple equation would be 100% reliable, but it's
not, because it's predicated on the assumption that the z-axis is
vertical and the z coordinate increases upward. It's just a matter
of time before you find that somebody has chosen the y-axis to be
vertical, or chosen z increasing downwards, or attached the z-axis
to an airplane that is maneuvering such that z is nowhere near
horizontal *or* vertical. Therefore you are much much better off
writing
ΔE = - m g • ΔR
where g is a vector (pointing down ... by definition of "down")
and R is the position vector. You have the option of expanding
R in terms of Rx, Ry, and Rz in some basis, but this is not
required. R is a thing unto itself, a physical and geometrical
thing, an arrow with a tip and a tail.
Some students who have seen too much computer science and not
enough physics think a vector is any old list of numbers, but
this is a Bad Idea. It is incomparably better to think of a
vector as a physical, geometrical object ... something with
magnitude and direction.