I have this timer circuit. Push and release vs push and hold give me different times
Your original circuit does not have a clear discharge path with known resistance and therefore its discharge time is difficult to assess. Chances are you have achieved 14 minutes through trial and error of different base resistors. Other factor that comes into consideration is sudden supply voltage drop when the transistor triggers current through the relay winding. This could explain why holding the switch and letting voltage stabilize gives a tad more charge to the 1000uF capacitor, and since discharge path is of unknown high impedance, this results in a higher overall time until transistors cut off.

Instead of letting the capacitor discharge through the Darlington base, create a known resistance discharge route by inserting a resistor in parallel with your capacitor. You should also add capacitors close to the 7812s like in the following schematic due to changes in load. Your LED resistor value should be decreased and base resistor value should be increased.







simulate this circuit &ndash; Schematic created using CircuitLab

To calculate the needed value of 390k, assume transistors cut off when the capacitor charge is lower than 1.4V, and that base current can be neglected. This means it needs to discharge from 12V to 1.4V, or 11.67% of initial voltage. This takes roughly 2.15 RC time constants, according to the following plot.



Let C be 1000uF, we have the following equation:

$$
2.15\times R \times 1000u=14min \times 60s/min
$$

R then equates 390.7k, or roughly 390k.