You could say you use Firefox, as the web browser is technically Firefox. I
agree that "web browser" on its own is weak branding, although Microsoft gets
by with "internet explorer" which is pretty much a synonym. I'm not too good
at naming web browsers, though, so I'm unfortunately of no help.
I know it's been suggested to use the GNU Icecat browser in place of the
unbranded Firefox several times before, but would it be possible to just
apply the Icecat branding to the current browser? I think it'd actually be
pretty cool if the other libre-firefox clones (e.g. gNewSense BurningDog) did
this too, in a sort of showing of unity. As far as I can tell, the
functionality between all of these browsers is roughly equivalent, and the
different brands can lead to confusion.
BTW gNewSense should only sound somewhat like "nuisance" if you say it
correctly - my understanding is that the "gNew" is pronounced like "GNU"
(hard G + "new") and "sense" is pronounced like "sense" as opposed to
"since." I happen to think it's a catchy name, myself, though.