I could be way off, but do you have an aftermarket alarm? I ask 'cuz my lady has an alarm in her CRX that doesn't sound off and whenever she drains the battery by leaving an accessory on, or when someone is working on her car and disconnects the battery, there's alot of confusion as to why her car wouldn't start because it goes to into 'valet' mode. But you can hear relays go off. It starts after you turn the key to first click and hit a tiny hidden button.

Like I said, probably off base, but thought it was funny how your symptoms (no clicking from starter; no evident system strain; etc.) are so similiar.

BTW, there were other instances when the alarm had gone off, but since it's a non-working siren, it drains the battery just enough to make the interior chimes to go a little goofy...!