The first holds up his pants. It corresponds to the celestial equator, which is determined by the terrestrial equator.

The second supports his six-gun and holster, and it is at a 23.5 degree angle to the equator. We call that the "obliquity of the ecliptic", or the tilt of the earth's rotational axis to its orbital path of revolution about the sun. The places where the ecliptic and equator cross, where the belts coincide at the equinoxes, are where the sun crosses the equator heading north in March and south in September.These occur traditionally (according to the astrologers) in the constellations of Aries and Libra, although precession has shifted them along to Pisces and Virgo.

The third belt keeps the pistolero's fanny pack securely lashed to his ass. It crosses his pants belt at an angle of about 60 degrees, and corresponds to the path in the sky of the Galactic Equator, or what we call the milky way.

Ecliptic and equator are related, the Sun and most of the planets and satellites rotate and revolve in roughly the same plane, with only a few exceptions, as you point out. This is no doubt an artifact of the rotation of the original nebula that formed the solar system. But the plane of revolution of the solar system, as well as the plane of rotation each of the stars, have nothing to do with the rotational motion of the galaxy. The rotations about their axes of the individual stars are totally uncorrelated and unrelated to the rotation of the galaxy. In fact, they appear to be completely random. Studies of binary stars and of stellar rotations indicate that the axes of rotation are distributed randomly, with no preference to the rotation of the galaxy.

As the solar system flies around the galaxy, the plane of the planets' revolution (the ecliptic) is tilted relative to the direction of travel by about 40 degrees.