Taking solar noon as being the time when the sun is at its highest elevation (altitude) in the sky for a given day, then your assumption of the LST (local solar time) is correct, as it is stated in the website you linked as being:
Twelve noon local solar time (LST) is defined as when the sun is highest in the sky. Local time (LT) usually varies from LST because of the eccentricity of the Earth's orbit, and because of human adjustments such as time zones and daylight saving.
This is something I have, in part, programmed into an app for a separate project (quite unrelated to yours).
You would have to equate the precedent quantities first (Local Standard Time Meridian, Equation of Time and Time Correction factors) as they have done on the website you linked (as these are reliant on the observer's position and time of observation), but to determine at what time solar noon occurred, the final step needs to have local solar time (LST) set to 12:00 (solar noon).
A worked exampled is given within the first 7 pages/slides of Sunlight and its Properties II.