I have had a Tivo for 2 years and the left green light always glows. I use a Harmony universal remote to control the switching on/off of the TV and TiVo.
On a second and muh more recent TiVo (again controlled by another Harmony remote), the green light will disappear when I switch off the TV.
I have reviewed the set up for both remotes and can see no significant differences.
I have checked the TiVo manuals that I got out of storage, but nowhere did I see a diagram explaining the lights.