Is the equinox Sun always at the same zodiac sign in the ecliptic belt?
As you said, the Sun takes a year to cycle the Zodiac, and it takes a year between two spring or autumn equinoxes. Both statements are true, but "year" in them means two different things. It takes a sidereal year to cycle the Zodiac, but it takes a tropic year to cycle between equinoxes. Since the tropic year is about 20 minutes shorter, the equinoctial points move across the Zodiac cycling it in about 25,000 years.

The calendar year we use in the Gregorian calendar is designed to approximate the tropic year to have it synchronised with seasons.

The reason for all it is that the Earth's axis is not fixed but its orientation slowly rotates.