I Prefer Design 1, the other one seems a bit 'lonely' being just one line and only having the C in a different font. On Design 2 I can also see where you cropped around the C in "Children" so you might want to fix that. In general though, I do quite like it.
I actually agree. The second one didn't really work because it removed pretty much everything unique from the old logo. Which proves the point in the thread I linked to and that inspired this, about why they didn't modify the existing idents with the new name in 1997. The reason I chose this one was for it's flat colours. I wouldn't be able to do the ones with 3D on them, or with busy backgrounds.
As for the C cropping, well, that's what happens when working with a JPG source I guess (the background is supposed to be one colour). And it's probably a VHS rip.
Why can nobody use the proper BBC logo? It's widely available. The version used here has square boxes and Gill Sans Regular lettering - the proper version has slightly rectangular boxes and custom lettering. I know it's a small thing, but it looks cheap and it isn't like the proper version is unavailable.