Now the first stage of the DT Timer is triggering when the battery is connected.

Add a 10K resistor to the gate of Q1 and ground in the motor timer circuit. My reasoning is the output of A1 maybe floating when power is first applied causing Q1 to be ON briefly causing the glitch.

 

Below is the LTspice simulation of AK's basic circuit with a few modifications:

R3 and R5 prevent excess current into the IC inputs from the capacitors if power is suddenly removed.

Edits:
Second Modification below:

Removed added diode D2, and instead moved R2's connection to the drain of the MOSFET to provide the reset function and generate the output pulse width.

The operation is as follows:
At power-up, R2-C2 sets the U1a-U1c latch (red trace) to its reset state (output transistor off).
The input pulse (green trace) flips the state of the U1a-U1c latch.
This starts to pull C1 low through R1 (blue trace)
When C1's voltage reaches the threshold voltage of U1d, its output goes high, turning on the MOSFET (yellow trace).
This low voltage at M1's drain also resets the latch through R2, causing U1a's output voltage to go back high.
This then raises C1's output voltage through D1, causing U1d's output voltage to go back low, turning off the MOSFET.
The value of R2 and C2 determines the time it takes U1a's output to reset high, and thus the output pulse width.

Note that the MOSFET must be a logic-level type MOSFET (max Vgs(th) <2V).

View attachment 251112

thanks for sharing information