This is not quite an answer but more than a simple comment. When I speak about distances I mean (angular) distances in the sky, when i speak about position I mean position in the sky. I adhere to terrestrial observer's point of view.
1, Starry sky (and everything on it, including Sun, Moon, planets) rotates around a point near Polaris (north celestial pole)
2, A given star (and everything which doesn't move relative to stars) gets to its maximum height above horizon at south (by definition). This is called culmination.
3, Interval between two consecutive culminations of the star is reasonably constant and is called sidereal day. Its length is approximately 23 hours and 56 minutes
4, Sun moves relative to stars along a well defined path, called ecliptic. This image shows starry sky along ecliptic on Apr 12, 2020 with Sun and planets. The ecliptic is the horizontal orange line in the middle.
5, Sun moves along ecliptic to the left and completes its journey in one year. The length of the ecliptic is 360 degrees, year has slightly more than 360 days so Sun moves by nearly one degree in one day.
At the end of May, for example, Sun every year passes between the most prominent groups of stars along the ecliptic, Pleiades and Hyades, both in Taurus. This is real photo from SOHO satellite.
6, When you measure culmination of Sun you must get more than sidereal day
7, Since, as many others noted in their comments and answers, our life is (still) controlled by daylight (Sun), we define day according to culmination of Sun, rather than stars. This is because solar (rather than sidereal) day has 24 hours.
6, If you define noon as time when real Sun is at south you get noon at different times for places at different geographical longitudes. This is serious problem when you need to coordinate say railway timetables. For this reason, time zones were introduced.
7, While sidereal day has (reasonably) constant length, solar day measured using real Sun has variable length during a year (motion of Sun along ecliptic is not uniform and different parts of ecliptic have different distance from celestial north pole). So the constant length 24 day is an abstraction - you must "observe" culmination of a fictional point called mean Sun.