3-SAT for variables appearing 3 times
There is no conflict between your references. The problem is that they have slightly different definitions of what 3-SAT is. You need to read the (original) theorems by Tovey very carefully.


  Let r,s-SAT denote the class of instances with exactly r variables per
  clause and at most s occurrences per variable. 
  
  [...]
  
  Theorem 2.1. Boolean satisfiability is NP-complete when restricted to instances with 2 or 3 variables per clause and at most 3 occurrences per variable. 
  
  [...]
  
  Theorem 2.4. Every instance of r,r-SAT is satisfiable.


The reason there's no conflict is that you need to allow 2 or 3 literals per clause to make 3-SAT $NP$-complete when restricted to at most 3 occurrences per variable. The trivially satisfiable bit only kicks in when you have exactly 3 literals per clause and at most 3 occurrences per variable.