For reapplying the new - I got one of those little squeegy things with the rubber blade and red plastic handle. Makes a really nice circular even corner bead the whole way.
I don't know a name for it other than "caulk corner applicator."
Like this one. - it says...
"After you've applied the new caulk, use the Caulk-Rite tool to remove the excess, and create a perfectly smooth, professional bead line! "