How long should you wait once you get the green light before moving forward?

Your statements lack a little detail.  If you had a green light, most times opposing traffic also have a green light.  Unless the other vehicle entered a left turn lane controlled by a signal, which is usually red if you have the green.  But if the other vehicle entered the intersection legally on a green light and you also had a green light, then the driver would be required to yield the right of way to any vehicle close enough to consititute an immediate hazard before making the left turn.  You have an obligation to ensure the intersection is clear as others have stated before you enter it on a green light.  Without a more detailed description of how that intersection is controlled by signals, it's difficult to determine if the other car was in the intersection legally.  Nonetheless, it is was, that driver still had an obligation to yield to you.  It's unfortunate that insurance companies place so much creedance in an opinion by a police officer investigating a collision because in the last ten years budget restraints have severely impacted police officers from obtaining advanced traffic accident investigation training, but they do the best they can.  Split seconds can make all of the difference in deciding who may have been most at fault.

Were there any independent witnesses?  Pleading your case to the insurance company generally doesn't do much good because their insurance investigators generally don't get involved unless there were serious injuries, high value loss or death.  You might want to consider going back to the police station that took your report and ask to discuss it with an accident supervisor, who approves accident reports.  They can always complete a supplemental report that can help clarify or even change the causal factors upon further investigation.  You might also revisit the accident scene, photograph the lights and the streets as you approach the intersection at the same time of day.  Also check that all of the signals are operating properly and that there no delays that would affect other traffic from entering the intersection faster or slower than you.