The equation of time is the sum of two effects: the uneven speed of an elliptical orbit (one wave per year) and the tilt of the axis (two waves per year). Together they set the width and lopsidedness of the figure-eight.
The up-and-down extent of the analemma comes from the Sun's declination, which swings ±23.44° because Earth's axis is tilted. The side-to-side extent comes from the equation of time: the Sun is up to 16 minutes fast or 14 minutes slow compared with an even clock.
In your sky plots where the Sun actually is (altitude and azimuth) at the same standard clock time every day of the year, so a camera fixed in one place would record this shape. Daylight saving is ignored on purpose.
Orbit playground draws the figure as seen from the equator at mean noon, for any orbit. With a circular orbit the two loops are equal; with no tilt the figure collapses to a horizontal line.