Values are for the selected place. The instrument names refer to those at Jaipur; the readings would be the same on any instrument of that type built for this latitude.
Local apparent solar time is time by the real Sun: 12:00 when the Sun crosses the local meridian. It is what any sundial, including the giant Vrihat Samrat Yantra at Jaipur, shows. It is computed here from the Sun's hour angle: solar time = 12 h + hour angle.
Clocks keep standard time for a reference meridian — 82.5° E for Indian Standard Time. Two corrections take you from sundial time to clock time: the longitude correction, 4 minutes for every degree the place lies west of the reference meridian (Jaipur, at 75.82° E, is 26.7 minutes behind), and the equation of time, the up-to-16-minute difference between the real Sun and an evenly moving mean Sun, caused by the tilt of Earth's axis and its elliptical orbit. Clock time = solar time + longitude correction − equation of time.
The Sun's position (declination, altitude, azimuth, hour angle) comes from the astronomy-engine library, with standard atmospheric refraction in the altitude. The Sun's rashi uses the sidereal zodiac with the Lahiri ayanamsa, and the lagna is the sidereal ascendant, as in the Lagna tool. The Jaipur coordinates used are 26.9248° N, 75.8246° E. Times are good to about a second; a real sundial is read to a few seconds at best on the Samrat Yantra and to minutes on smaller dials.