I know it's pretty much pointless to try and show you why your line of thought is wrong so I was brief. But I'll try anyway...
A logo doesnt have to be so squarish which is your line of thought For whatever purposes that is for (I think it has something to do with the positioning of the logo on the site, with a longish logo not fitting into the site? Something like that?)
The hat one is by far the best design, it was the most popular design in the previous discussion of logos and somehow we ended up with that ...thing in the top right hand side of our screens.
I don't see why there has to be a banner and why there has to be a logo. I think using the same design for both is perfectly fine. The hat is the little something special that makes this banner a logo
Edit: If you do want a squarish 'logo' for whatever weird reason, maybe you can have VCE on top of NOTES and then on top of all that you can have the hat. Soz but paper looks a bit uninspiring . no offence ice