non-inverting schmitt trigger op amp - replacing a zener diode on the input
The Zener, in combination with the 1K resistor, is forming a stable voltage source for the comparator to compare against.  If your 12V supply is stable enough, you can divide that down and use that for reference instead.

But not the way you have shown.  The Thévenin impedance of the original circuit is ~0 + 10k, or 10k ohms, but the Thévenin of yours is 10k || 1k, or 909 ohms.  So the original circuit will have much more hysteresis than yours.

The CircuitLab simulator should be sufficient to show you the difference between the two circuits.  In addition, you can use it to get a reasonably-drawn schematic.