3 way switch mystery
Congratulations!  You just found a Carter 3-way!

This configuration switches each "side" of the lamp between hot and neutral, as depicted in the illustration below:


(source: Wtshymanski/Wikipedia)

However, due to the switch in the neutral wire, it was possible for the shell of the lampholder to be hot even if it was switched off, leading to a shock hazard!  As a result, this configuration was banned by the NEC and remains forbidden to this day.  You were correct in removing the Carter 3-way from service.

The second circuit feeding the Carter 3-way was also a rather egregrious Code violation (300.3(B)/310.10 if nothing else), but your work has cured it as well.  (While the current configuration isn't completely Code-conformant, you've gotten as close as you reasonably can without running new wire.)