Actually Gothic is this adjective of Goth, name of the original inhabitants of this part of Europe (current Germany and Austria). Note that the spelling and pronunciation is the same in every language.
Later Gothic was the name of the architecture that mainly spread out during high middle ages, that means the centuries before the Italian Renaissance; it is often opposed to Roman architecture, especially for churches.
The term has been continually used in art such as novel or music to express things coming from Germany till today, where the Gothic style is often synonymous of wearing black clothes, black make up on artificially white skin, piercing, etc. and of course, the last but not the least, adhering to the gloomy (yet harmless) Gothic rock (ex. the Cure).
I think that style comes from Central Europe and I'm pretty sure US has been reached, but I don't know wheter it has spread out to Japan or Taiwan.
Back the thread, whether 哥德 or 哥特 is chosen, just keep in mind that Gothic = German is too simplistic, especially today where its influence encompasses all the Western world, and Gothic as the name of an art school was multimillenium old, even at Goethe's era. Finally, the Gothic as an adjective is the Goth style, 哥特式 or 哥德式, I personnally they are both acceptable.