First, there is actually nothing intrinsically wrong with circular definitions, at least they are internally consistent. There will usually be either some circular definitions or some undefined terms. However, in science the basic definitions are usually not circular, they are usually operational. This means that they are defined by some sort of experimental measurement.
This is the case with mass. If you had read the rest of the wiki article you cited then you would have seen that the first sentence which you quoted was intended to be a description, not a definition. The definition was in the section labeled “Definitions” where it says “There are a number of ways mass can be measured or operationally defined:”. It then proceeds to give operational definitions for the various concepts known as mass.
More simply, mass can be operationally defined as the quantity measured by a scale. It can also be theoretically defined as the quantity $m$ in the expression $m^2 c^2=E^2/c^2-p^2$. This is a nice theoretical definition because it allows you to determine the mass of things that you cannot put on a scale.