What Does "Set SCE To AUX" Mean Anyway - Apollo 12's Lightning Strike Explained

It's an often told story about Apollo 12 getting struck by lightning, crippling the command module in flight and John Aaron in mission control figuring out the exact switch to flip to fix the stricken vessel and save the day. It's an understandably popular story, but the details are never quite right, so this is my attempt to explain what exactly went wrong and what problems SCE-to-AUX fixed.

Here's a few of the references I used to understand what happened:
Apollo 12 Flight Journal
https://history.nasa.gov/afj/ap12fj/0...
Apollo 12 Mission Control
https://history.nasa.gov/alsj/a12/A12...
Apollo 12 Saturn V Performance
https://history.nasa.gov/afj/ap12fj/p...
John Aaron Interview for NASA Oral history Project
https://historycollection.jsc.nasa.go...
Apollo 12 Technical Debrief
https://www.hq.nasa.gov/alsj/a12/a12-...
Apollo Command Module User Manual - Telecoms Subsystem
http://apollo.josefsipek.net/ApolloPr...
Analysis of Apollo 12 Lightning Incident
https://spaceflight.nasa.gov/outreach...
Scott Schneeweis Collection of Apollo Hardware:
http://www.spaceaholic.com/index.php/...