Step 1: Check on the Ignition Switch Recall.I'm bumping this because I'm having the same issue now. Trying to get everything else on my car worked out while I wait for this Type-S swap to get here. The engine still runs alright just rod knocks like a bitch, I don't drive it but it should still crank no problems.
My car is doing the same thing as the OP, just plain will not even attempt to start. I'm doubting its the main relay since it does not try to start at all. It's a 2001, which is not eligible for the ignition switch recall (and I would think the issue would have been addressed from the factory). I have checked the clutch switch and it appears to be working fine mechanically. I do not currently have a voltage tester, as mine died.
My question is how do I determine which of the several possibilities is actually causing my issue, before I start randomly swapping out parts? I'll be really pissed if I finally get this Type-S engine in and then it won't start because of this issue...
Step 2: Push-start the car anytime the ignition doesn't work after several attempts.
Step 3: Fix the loose wire.