Im not sure right now if the relay coil is grounded and 12v switched to it or the other way around. Im assuming one end is grounded, and you got ground through the relay coil and the other end at 12v gave you current flow to light the light in the tester. When you turned the ignition switch Im not sure why it would get brighter in that case unless the switch is grounding one end of the relay coil.
But it certainly cant provide a ground if one end of the tester is at 12v and t he other end is on the relay coil. Since some current flow was provided through the tester into the engine warning light, I would have to assume the engine warning light has its own ground to the battery. And most likely the coil wire labeled `i` gets 12v from the ignition switch. For some reason when you turned the ignition it presented the connection labeled `i` with a contact to ground of less resistance than that of the relay coil, causing your test light to get brighter. A very low resistance at the ignition switch doesnt sound good, but it could be just normal and not really a short. Just lower resistance than the relay coil. Resistors in parallel divide.
It sounds like no 12v is getting to ignition switch. Grounds should be ok. Check for 12v at the switch. I wouldnt even suspect the switch right now.