On the coils, I agree that the instantaneous current is what gets the work done, but in practical application they only need to dissipate the average current. In the application of a switch machine coil there is no need to worry about the peak current here, though one could go into the the details all day, but what it comes down to is that the coils can handle being over-driven much better than the LEDs can, and likely the coils are designed to support a 18VAC input anyway.
On the .707 vs. .637 subject, I've seen both numbers used for the same thing. I prefer .707 for a couple reasons. There seems not to be any debate over using 1.414 to convert the RMS voltage of a sinusoidal wave to its peak to peak voltage. At least I can't recall any other other number but the square root of 2 being used anywhere. As such, it makes more sense to me to use the reciprocal of SqRt 2 (1.4141) when reversing the calculation. That is .707 times peak equals RMS of the sine wave.
Take for example a pure sine wave AC voltage of 18.000 VAC RMS. One can easily discover the peak to peak voltage of this source by multiplying by 1.4141 (sqrt 2 rounded off). That equals 25.4538 volts peak to peak. Now when one has the peak to peak voltage and wants to determine the RMS voltage there should be some similar simple conversion. That is where .707 or .637 comes in.
Say you know you have a peak to peak voltage of 25.4538 volts and you want to know what the RMS value is. Now from the previous example we know that this is 18 VRMS but what do we use if we did not know? Well some folks like .637. So, 25.4538 * .637 gives us about 16.214 volts RMS. On the other hand, 25.4538 * .707 gives us 17.996 volts RMS One of these is clearly much closer to the actual value.
Now I am open to the idea that the square root of 2 may be the number that is wrong here, but I've never seen anything to suggest that before.
All the theory aside, Junior, use a 1.5k Ohm resistor in series with your LEDs and they should work well at voltages of 10-20 VAC on the input side. Assuming a typical LED designed for maximum output at 20mA of current.
JGL
