Starter Motor not pressing instantly

2001 pejo 206 with my car, about 5 km, I stopped the car for parking.This incident happened at night. In my trials, after two or two hours intervals, the first time the vehicle took off, the vehicle is automatic and I am about to go on a long journey to the holiday, but I am afraid if there is any problem in the vehicle.

If you are constantly using the vehicle at short distances, the starter motors that are likely to be out of power may instantly draw current at very high amps.