It's not just DST and distance from your time zone's central meridian that affects the time of true noon (aka apparent noon) i.e., the time when the Sun is highest in the sky that day, when the length of the shadow of a vertical pole is at its minimum for that day. The

average time period from one true noon to the next is 24 hours, but the actual time period between successive noons varies throughout the year, mostly due to the obliquity of the ecliptic and the eccentricity of the Earth's orbit.

To keep things simple, modern timekeeping uses days of fixed length. The difference between local mean noon and local apparent noon is called

the equation of time.

From that Wikipedia link:

The equation of time — above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow.

So the difference between clock time noon and local apparent noon can be more than a quarter of an hour, even for someone located right on their timezone's central meridian with no DST.