Why 纹 is the wrong character to write for tattoo?
They are purists. In the words of Steven Pinker:


  ...also known as sticklers, pedants, peevers, snobs, snoots, nitpickers,
  traditionalists, language police, usage nannies, grammar Nazis, and
  the Gotcha! Gang.


According to this article, 纹身 is accepted by a newer version of 《现代汉语词典》 as an alternative form of 文身.