It seems to me that if everyone is infected so that even a natural death would turn a human into a zombie, why is there concern over non-lethal zombie bites? For example, the need to cut off Herschel's leg. Does the bite itself, no matter how insignificant, cause death?
Everybody is already infected and will turn into a zombie upon their death, as such, to avoid turning into a zombie, you have to avoid death in general.
A bite from the walking dead can very quickly turn infected and gangrenous, which without proper medical attention will result in a painful death. As such there isn't really such a thing as a "non-lethal zombie bite" because you can't be certain that it will not cause a festering infection.
The canon explanation in The Walking Dead is that "Everyone is infected with a dormant/passive version of the virus."
Upon the host's death, the virus activates, transforming the host into a walker.
When bit by a walker, It transfers some of the live/active virus into the victim. This virus stays active, slowly killing the host through the process of making them a walker.
This is explained & shown in the final Season 1 episode of the TV series, TS-19:
What isn't explained in the video, but later in the scene, is that the Test Subject had been bitten, and had volunteered to be monitored so the CDC might find a cure.
The zombie bite contains a more powerful, or fully developed form of the virus. It spreads from the bite area and eventually kills the victim. The only way to prevent it from turning the victim into a zombie is to remove the infected area from the body. A bite to the arm or the leg would require that you amputate the bitten limb. If you get bitten on the neck or the shoulder then there isn't much you can do. Just go somewhere away from everyone else.
Yes, the change happens a few hours after one is bitten or a few hours after when one dies whichever comes first.
Another way of looking at it is that zombie bites are highly venomous and normally kill the victim within a few hours and they then turn.