OK with that working, now this should get the trigger working correctly I think:
The 560K/ 330K (R3/R4) voltage divider should provide around 4.4 volts with no voltage from the coil. The 555 trigger voltage should be 1/3 of 12V or about 3.6V so if the coil manages to produce at least -1 V (and the larger value resistors should let it do that I hope, if not we can increase the size of R3 and R4) that should trigger the 555 and you should be able to stretch the pulse so you get both enough light without overloading the LEDs and enough time on the output pulse to ignore the second trigger you were seeing. You will need to adjust the R2/C3 values to get the appropriate length of pulse for what you need,
Peter