A more detailed way to think about it is this:
There are three* component motions to the sun's apparent motion across the sky. One of them manifests itself as the sun's precession along the ecliptic, the other two manifest themselves as an apparent shift of the sun and ecliptic plane (together).
In components, the solar motions are these:
1) A regular precession of about 1 degree from west to east per sidereal day, caused by the Earth's orbit about the sun. This is what causes the stars to shift form night to night, and why the winter sky is different than the summer sky.
For time keeping purposes, we have defined a 'day' to be a solar day (The time the sun progresses through 360 degrees), instead of a sidereal day (The time the fixed stars progress through 360 degrees.) Consequentially, this 1-degree of precession is not seen in the sun. If terrestrial time were kept with respect to the stars, this effect would manifest itself by shifting sunrise and sunset by about 4 minutes a day as the year progressed.
2) A vertical shift to the North or South. This is caused by the Earth's axial tilt relative to the ecliptic, and manifests itself as the vertical 'height' of the analemma. This effect is also used to define the Equinoxes and Solstice.
Equinoxes are defined as occurring when the sun occupies one of the two fixed locations in the sky where the great circle of the ecliptic crosses the projected great circle of Earth's equator.
3) A minuscule shift to the east or west. This is a minuscule effect which results from the Earth's orbit not being perfectly circular. Consequentially, Earth's orbital velocity is not perfectly constant, compared to our rotation rate, which is more or less constant. This effect manifests itself as the 'width' of the analemma, and also as +/- 16-minute deviation from 'solar time' to 'true time.'
You are right that the sun traces out a figure 8 (analemma) over a course of a year, but this is due to terrestrial effects (3) and (2), and has no effect on the shape of the ecliptic.
Only effect (1) causes the sun to trace out the ecliptic.
*To be perfectly fair, there are actually four. The fourth apparent motion of the sun is caused by the precession of the equinoxes, which results from the Earth's axis slowly shifting orientation in a 26,000-year cycle. But this effect is insignificant compared to those listed above.