Security system?

Hello,
Every morning I have to keep my car in the key "on" position for about fifteen to twenty minutes. During this time, it looks like it does some circuit check as all lights show up and go off, but the check engine light stays on. After the waiting period of fifteen to twenty minutes, check engine light goes off. When the light goes off, there is a sound which kind of feels like the circuit is complete and the current is now flowing and there is a small click sound. Now if I crank my engine, it will start immediately, but it will not start if I try to crank it during the waiting period when the check engine light is on.
Then all day it starts up almost immediately as the check engine light (or initial circuit test) is completed immediately in key "on" position.
I have just replaced a new battery but it did not help. I would very much like to know why this happens and what should I do to fix it?
Thanks.


First image is when car will not start after cranking. Second image is when the car starts after all lights are gone.