The "New" Zodiac and the Earth's Spin

The latest rage in popular astronomy seems to be the realization that the Sun may now move through 13 constellations instead of 12. I haven't personally checked this, so I'm going to take their word for it. What's bothering me is something else that's being repeated in connection--that this is "due to shifts in the earth's rotation and orbit", or more flagrantly, that "since the zodiac periods were established millennia ago, the moon's gravitational pull has made the Earth 'wobble' around its axis in a process called precession".
Someone (I guess, me) needs to make it clear that any shift in the apparent path of the Sun through the background constellations can only be caused by a change in the inclination of the Earth's orbit (possibly as a result of the precession of the Earth's slightly inclined orbit around the Sun), not the precession of the Earth's axis of spin.

The reason the Sun appears to move through background constellations is because we're moving around the Sun (see figure to left). From the perspective of Earth (blue dots), the Sun (orange) appears in front of a different set of stars at different times of year, when the Earth is at different positions in its orbit. The project of the Sun from the perspective of the Earth is shown with dashed lines.

Now the thing is, that dashed line from the Earth through the Sun doesn't depend at all on the Earth's spin. To see this, imagine you remove the Earth from one of those locations and instead place yourself floating in space. You could turn any which way you want--upside-down, rightside-up, twisted alley-oop--the Sun is going to be in the same place relative to the stars behind. Similarly, the Earth's spin (and any precession in that spin axis) can change the orientation of the Earth, but makes no difference in what stars the Sun appears in front of.

What actually can change where the Sun appears relative to background stars is the physical location of the Earth.
This is a change in the Earth's orbit, not spin. One way to change where the Sun appears relative to background constellations is to move the Earth to different places in its elliptical orbit. This will move the Sun through the standard set of zodiacal constellations. In order to move the Sun out of the normal zodiacal progression, you need to move the Earth up or down. The way this can happen is if the plane of the Earth's orbit precesses around another axis, so that the "high" point in the orbit moves slowly around the Sun.

So to clarify, if the Sun appears to move through a new constellation, it is because the Earth's orbit around the Sun has changed, not because the Earth's spin has changed.