Sorry, I am a mathematician and all those funny little logical connectives have very specific and precise meanings in my mind. I was not aware that "ergo" was in the same boat as the ever so sloppy "or" in being so imprecise in common usage.
As for you example of turning off the lights I parse the two sentences differently. In the case of 'so' you are not making a claim to logical necessity. An explanation based on simple correlation is acceptable here. In general these things are done in this order, but it is not necessary that they are. When using "ergo" the logical dependence is required.
Another potential explanation is that all "ergo"s are "so"s but not all "so"s "ergo"s.