The term anti-matter was not invented by particle physicists, but by science fiction authors, so there is no point in studying the Standard Model to see how it defines the term. Moreover, particle physics does not even seek to define the meaning of the term matter, without which there can be no logical meaning to its opposite, anti-matter.
Einstein proposed that particle physics should be based on the energy equivalence principle: thus matter as a concept has no meaning, since particles consist entirely of energy, albeit in a state of confinement. Release that energy from its confined state, and you have a Hiroshima-level of energy liberation.
The classical science fiction depiction of anti-matter suggests some very similar large type of explosion, if matter and anti-matter meet. But it is more logical to regard a particle and its anti-particle as neutralising one another. A particle confines a huge quantum of energy, but has a relatively tiny electric charge: if it encounters an anti-particle the two will behave in essence like any pair of charged particles: +1 applied to -1 will sum to zero.
What we ought to be considering is the mechanism by which a particle and its anti-particle differ from one another. If the difference turns out to be a question of spin, for example, one could see how two opposed particles (i.e. having opposite spin) could meet and in effect neutralise one another, without necessarily disturbing the confinement of the energy bound up within them.
It is evident that most members of the particle zoo do not lend themselves to a neat division between matter and anti-matter. What modern physics needs is to avoid any such old-fashioned terms, which belong firmly to classical 19th Century thinking. The notion that the universe can be divided into two opposed camps flies in the face of everything we have discovered in the past hundred years: the electron has an opposite charge to the proton, but no scientist proposes to label the electron as anti-matter. And when the electron and proton meet, they co-exist - they do not mutually annihiliate, there is no explosion; they sit together, and merely cancel each other out (in terms of their opposed electric charge).
We cannot yet see a quark, or an electron: they are billions of times smaller than the best current visual aid - the electron microscope - can magnify. But only by seeing them will we determine why a quark differs from an anti-quark.
What we do know is that a high energy x-ray laser is capable of splitting an electron into three distinct particles: one carrying its charge, one its spin, and the other its orbital motion. So we should be careful not to be "sure" that an electron represents anti-matter, when we are not even sure that we really understand what an electron is. Clearly, it is not a particle: it is at best a combination of three particles; just as a proton is not a particle, but merely three quarks in combination.
Now that we have the beginnings of a means for studying the particle which gives the electron charge, we are one step nearer to an understanding of what causes charge, which can help us understand what gives charge to a quark, and why many particles in the particle zoo don't have charge at all.
Perhaps that may explain why some quarks can anomolously lack charge, causing their proton to exhibit negative charge (so-called anti-matter), or how an electron can exhibit positive charge (again, so-called anti-matter).
We are assured by no less an authority than Einstein that so-called matter is really a combination of energy and the square of the speed of light: so we know that particles are really a field of energy, not a solid object. It is a small step to then suggest that particles may be sub-divisible -
as the electron is now seen to be.
The quark must also be divisible into sub-particles, one of which carries its charge: most likely the same sub-particle as carries the electron's charge. Some packet of energy, responsible for what, at a larger scale, we perceive as a difference between matter and anti-matter.