Firstly off, the focal length of the lens itself never changes. It is a fundamental property of the lens.
A Canon 2x tele-extender adds 58mm between the objective lens and the sensor plane. So by definition, the focal length of a 200mm lens is “changed” into a 258mm lens.
This 2x tele-extender magnifies the central portion of the frame 2 times so it gives the same picture as a 400mm lens would.
With my 1.4X ii I get an "effective" 420mm using it on a 300mm lens and an "effective" 700mm using it on a 500mm lens.
And with my 2X ii I get an "effective" 600mm when using it on a 300mm lens and an "effective" 1000mm using it on the 500mm.
All this is per the EXIF data. (example: http://www.flickr.com/photos/amfoto1/6865952788/in/set-72157622481646019)
Rarely use either TC on zooms (too much of a hit to IQ for my tastes), but when I do the results are the same... EXIF shows my 70-200 + 1..4X "becomes" a 112- 280, for example
You are probably using a Canon brand converter on a Canon brand lens. I doubt the OP is.
I totally agree the penalty for using one is not worth it. Unless it is just the only way you can get your shot.
If you have to use one whatever the lens brand is, is generally the best choice, however.
Why am I only getting a focal length of 365 when I use my 2xiii with my 70-200?
Unless you have tested it, making sure you are zoomed out all the way, I'ld say it's likely you're moving the zoom ring slightly as you make the photo. I frequently think I'm at maximum focal length only to get back to the computer and see that I'm not quite out there.