I would say that, perhaps, a more suitably general, conceptual, "physics"-related definition of mass is that mass is a numerical parameter which characterizes the responsivity of an object to an applied force. When you apply a force to an object, it responds by changing how it is moving, whether that is to speed up, slow down, change direction, or some combination thereof.
To understand this notion of mass better, consider the following experiment. Suppose we start with two different objects. We will take them as initially being at rest, to make things simple. Then we subject them to equal forces that do not vary in time with regard to either strength or the direction from which they are applied. What we will find, of course, is that the objects begin to speed up. We can then make a graph of their speed versus time elapsed under force. This is called an "acceleration profile".
If we repeat the experiment with the same objects and forces many times, we will also find that each time, the acceleration profiles they give are always the same. However, with regard to each other, the two objects will, generally speaking, have different acceleration profiles. In other words, there is some property of objects, which differs from one to another, that causes them to respond differently to identical forces.
We can then explore the nature of this difference with further experiments. Suppose, say, we now take the two objects and glue them together to form one larger object, and repeat the same experiment, with the same force that we applied to each object individually, only now being applied to the combined object. What we will find is that this combined object has a shallower, i.e. slower, acceleration profile. Also, suppose we now, say, change the shape of the object, i.e. suppose we put a dent in one (but not break off any material, so think our objects as made of a malleable metal). We will find that the acceleration profile remains unchanged.
Through many experiments like these, we can conclude that we can characterize the acceleration profile that will be had by any given object for a given force by a single number we can assign to that object, and also, that this number behaves in a sense like an "amount of matter" in that if we pile more matter together, the number should be made larger. We call this number the object's "mass".
Now, when it comes to elementary particles, the key is remembering the more fundamental definition of mass as dealing with responsivity to force, and not the derived fact that it behaves as though it were also an "amount of matter" when we consider macroscopic objects which are the aggregation of many, many elementary particles. There is no real sense in which we can say they have more or less "matter" - instead, the mass is an intrinsic property that tells us how they respond or not to an applied force.
Relativity does not change the basic notion: what it changes is the nature of the acceleration profile - in particular, in non-relativistic Newtonian mechanics, this profile is always linear, i.e. the speed increases at a steady rate over time, but in relativistic Newtonian mechanics, speed rises to a plateau: the so-called "speed of light", $c$. Different objects still can have different acceleration profiles, and thus we can still distinguish a mass parameter, at least conceptually. Mass isn't about the shape of the profile, rather, it is about what discriminates between them. In particular, larger masses will rise in speed more slowly initially as in non-relativistic mechanics, but also will plateau later. Smaller masses will behave in the opposite fashion, speeding up faster and plateauing sooner.