FYI, Toyota use ground switching for most of their electrics. This means that the positive terminal of the boot light will always have constant battery voltage present.
So get yourself a multimeter and with no load on the boot light circuit (ie. unplug your LED's etc) and with your boot latch locked, measure the resistance between the negative terminal of the boot light socket and ground. You can use the boot latch as a ground point, otherwise an amplifier body ground will do. If you measure any form of resistance. Ideally it should be an infinite reading. It won't take much contact with ground somewhere to mess things around. If you are seeing a few megaohms between the negative terminal and ground, then this is all it takes to get the LED's to give a dim glow. Obviously the less the resistance, the brighter your LED's are going to light up.
If you are getting an infinite reading, I would then be leaving your LED's the way they are and try to disconnect the connection to the light socket. Then use your multimeter to test between the negative terminal of the LED's to ground. Same deal as above. If you get any reading there that isn't infinite, I would make sure your LED's are completely isolated from the body of the car. If you get an infinite reading there, then I would be sussing out your connection to the lamp socket itself.
Those pictures... despite being massively sized, do not really give me a clear idea of their construction. Those LED's don't appear to be the ones that are completely sealed in silicon. They do look like there is the potential for part of it to ground itself with the body of the car. If you ask me, I'd reckon the issue lies in that.
This is why I use and recommend LED strips that are completely sealed when fixing them to metal surfaces of the car, or to where they may potentially contact metal surfaces. The ones I use look like this: