Is there a term for "path of the Sun at a given point of the Earth"
Merriam-Webster defines diurnal arc:

the portion of the diurnal circle of a celestial body that is above the observer's horizon

and diurnal circle:

the apparent circle or parallel of declination described by a celestial body in consequence of the earth's rotation

The NOAA Solar Calculator does not implement the exact feature in question but can show the Sun's azimuth for any given location, date, and time.