[Andreas]

A Mod (short for modification) could be just anything - if I replace the engine of my car with a flux compensator this is a modification, but surely not a Hack.
Now where does the word Hack (in terms of computer programming) originally come from:

Originally Posted by Wikipedia

The initial hacker community at MIT [...] A "hack" now meant a quick fix to a computer program problem, as in "That hack you made last night to the editor is working well". A hacker came into the lexicon as meaning one who hacks, using this definition.

http://en.wikipedia.org/wiki/Hack_(technology_slang)
Therefore Hack is, technically speaking, the correct word.

A Style defines the visual appearance of smth., for example every human being has it's own Style in regards of clothing, etc.
Regarding websites, this can be done with Cascading Style Sheets (CSS), which was first introduced in 1994.
AFAIK, the term Skin was used at first for customizing the look and feel/apearance of Software by WinAMP in the late 1990s.
A Skin consists of Graphics (and layout definitions, etc.), while a (vBulletin) Style does not necessarily have to come with new images.
Therefore hte Term Style is correct too