I think the most useful thing to start with is that the Mozilla Firefox branding is copyrighted by Mozilla, the branding is NOT under a licence that allows use of it without explicit permission and that the more ideologically pure distros didn't like that. All three cases mentioned on the Icecat page are simply re-logoed/branded versions of Firefox, done by different teams - from the Icecat webpage
The gNewSense BurningDog browser and the Debian IceWeasel browser are
similarly derived from Firefox, also with the intent of being free
software. Technically, however, these projects are maintained entirely
independently of IceCat. (Previously, this GNU browser project was also
named IceWeasel, but that proved confusing.)
Iceweasel is probably also packaged by Debian for use in their distro, as would be Burning Dog (which itself is part of a 'pure' FOSS Ubuntu spinoff). They're all Firefox code, and should work the same way as firefox, but are done by different groups of people for the sake of total FOSS compliance.
To answer the specific parts of this questions - yes they are different projects, they are just rebrandings of Firefox, Debian only has Iceweasel packages since they run the Iceweasel project, and as far as I can tell, Icecat changed its name from Iceweasel to avoid confusion and none of the spinoffs have merged. A great anology would be Centos and Redhat - exact same situation with Centos compiling redhat's packages after stripping out the branding.
EDIT: As of 2016 Iceweasel is no more, and the firefox branding is used by debian again - As per this raspberry pi stackexchange post by angussydney. Pop went the weasel I guess.