Car starts without key!

I would repeat the test, making absolutely certain you did it right, and did you? unhook the switch connector?

What you should have "in series" is this:

Your battery, positive unhooked, battery ground connected.

Your test circuit should be

battery ground---to battery---battery positive -- to light--light to red switch wire.

Then you should have a grounded clip lead which you can move to each other color of the switch, checking switch positions.

Sounds to me like the switch is bad, but this doesn't really explain why the starter was running, unless the switch is all broken inside, and when you removed it things moved around in there.