I just got my Raspberry disconnected from a 1 meter USB cable and then connect an extension USB cable (about 2 meters). Before replacing the cable the Raspberry functioned normally, but after replacing the cable, the green LED flashes and nothing happens. I tried 2 chargers: one 1A and through the gate with a second outlet with Raspberry 2A but this does not have any effect.
If you want to check the current draw over USB, then there are many types of USB voltmeter/ammeters available on eBay, for example:
This may help you track down the problem. However, as joan suggests in the comments, if you must use a 2 meter cable then get a thicker USB cable, so that there is less power drop.
I think you have pretty much done the troubleshooting yourself, the problem is most likely in the cable. It could be a really low quality one, since moving from 1A to 2A doesn't change a thing. Maybe the cable is even broken.
You could try and see of the extension cable works with connecting i.e. your phone to a PC (if it's that kind of cable).
The longer the cable the less efficient it will be, so might as well go for a proper one in your case!