Umm, It looks to me like your Targus Adapter should have worked, 8A > 3.5A and 70W > 45W (These are really the same thing as Watts = Volts x Amps.)
One thing to check though is the polariity. Sometime the center pin on a PSU plug is +ve sometime -ve I believe there is an unwritten rule the center should be +ve but not everyone follows it.
Also I would be a little wary of just plugging a DT 366 directly into the Cigarette adapter, some crappy cars dont isolate it correctly dropping the voltage allot when the car starts and if you get a fault in your cars charging circuite you could send 40+ volts to your WebDT. To do it correctly you would need to buy or build something like this:http://eegeek.net/content/view/20/31/
But this probably amounts to what you have in your Targus adapter.
An alternative simple solution might be to get a small DC-AC inverter and just use the power supply that came with your WebDT. Beneficent but should work.