Megan, I think you're right about it being a UK/US language issue. What I would call a Fin, the US would probably call a vertical stabiliser (probably with a z!). They tend to call our Tailplane a Horizontal Stabiliser.
It is however also a function of what the surface controls. The Phantom 'horizontal stabiliser' was indeed an all-moving tail but only controlled pitch, so it's not a Taileron. The Harrier had similar. The Bucaneer also had an all-moving tailplane (T tail) but also had an adjustable flap/tab on the trailing edge. I believe this tab was deflected up to counter the large nose-down pitch associated with lowering the flaps. (In fact the ailerons drooped at the same time, all indicated in the front seat with wonderfully named 'cheese' indicators! They had to all move together!)
The Gnat had elevators, but these were usually locked to the all-moving tailplane - the whole thing moving to control pitch. If needed, (Hyd failure), the elevators could be unlocked from the tailplane - the latter now being fixed or trimmed to a specific setting with the elevators now being controlled by the stick.
Jaguar and Tornado both have 2 one-piece surfaces that control roll and pitch - hence as Exrigger mentioned - we call them Tailerons.
At one stage the US called combined elevators/ailerons on the (Gutless) Cutlass 'ailevators'. I much prefer our Elevons!
Back to the Buc, I'm not sure if it's drooping ailerons were called Flaperons.
The worst name has got to be 'Speedbrakes'. I much prefer our UK name of Airbrakes.
On some types even the trusty throttle has picked up a dreadful Americanism of 'Thrust lever' yuk!