I have seen a remark saying "we usually say that a turing machine accepts/rejects a string, while it decides a language" Is this correct?
As I have also seen places where we mention a Turing machine "accepting a language"
See comment on OP's answer here, then the answer by Jan Hudec : What is the difference between a TM accepting and deciding a language?
I have also seen the definition of total/decider to mean, the Turing machine halts on all inputs. Is this all inputs in the language the Turing Machine is defined over?