Does the single character 神 refer to "God" (the unique God, such as in e.g. Christianity) or just "god" or "deity"?
It depends on your religion.

Traditionally,「神」is just a supernatural being, applied to anything from mountain gods to river gods to the sky god.

Ancient China only ever had a "Highest God"「上帝」, not a "Unique God" like Christianity/Islam/Judaism.「上帝」was never said to do things like create the universe. If you're looking for special vocabulary for a "Unique God" in the Chinese language, you won't find one, because that is not really a Chinese concept (nor a concept in most other ancient languages, which were all polytheistic, if hierarchical).

All Chinese-origin words that you see which purportedly refers to the "Unique God" are strictly a repurposement of existing Chinese vocabulary by Christians (or whatever other religion). Specifically,


「神」and「上帝」are used by Protestant Christians in an attempt to connect Christianity to some form of native Chinese belief. Specifically,「上帝」is explicitly used to connect the Christian's God to the Ancient Chinese Shangdi.
「上帝」is avoided by Catholic Christians, who overwhelmingly uses the term「天主」. Apparently, this is due to the exact same reason as Protestant Christians but in reverse: a previous pope wanted to explicitly avoid the Christian's God with Chinese Shangdi, because Shangdi was just part of a polytheistic religion.


Ironically and rather unfortunately for Catholics,「天主」was the name of an ancient non-unique deity in the State of Qi, and was only one deity among eight.





In casual language,「神」is a kind of slang meaning amazing or awesome, in the same manner that god is used as a word for someone who is really good at something.


  「你打球好神！簡直比NBA球星還厲害！」
  
  Your [basket]ball play is amazing! Even better than NBA stars!


Unless someone explicitly states that they meant the Christians' God, I would rather think that the tattoos of「神」mean god as in "I'm a God/I'm awesome", etc.