Toyota echo immobilizer or anything else. Guys help me out.

[humayun.chughtai.7]

I am facing a weird problem with toyota Echo 2001 hatchback (same as vitz) since I bought it.
- Engine refuses to shut down, while all the electricals turns off, upon turning the ignition swith to ACC/LOCK position. But this only happens when engine is cold & blue coolant temperature light is on the board. When blue light goes off, everything seems normal.
I was guessing that there was some remote start kit installed by previous owner, and it had some wiring malfunction. Or any other aftermarket accessory.

I inspected all the wiring including under dash board & engine bay but everything is in standard condition. This is where I came to know that 2 wires of one of the ECU connector are attached (connecting with each other after peeling off the insulation) with each other. Here is picture of ECU pinouts for better understanding. (Wires circled in Red)

When I disconnected them from each other, car refuses to start while starter was cranking & there was NO Check engine light in the odometer. I also noticed that the security light (which refers to immobilizer) also didn't lights up due to this.
So I am concluding that my car has an immobilizer, someone had tried to bypass it by directly connecting the 2 wires. Plz correct me, if I am wrong.
Plz help me in this regard, what to do now ? how to solve this matter bcz after a short trip of 3-5 kilometers, it feels very weird when u want to turn off ur vehicle and it refuses to shut down.