"Ill" and "sick" has different meanings in British and American as far as I know. So from the American point of view I think "hasta" is "sick". "Ill" can be translated as "fena" both in the sense like "an ill behaviour" "fena bir davranış", or like in the sense "I am ill" as "fenayım". You can also use the word "rahatsız" which literally means "unconfortable".
I think what they are talking about is the difference between ill "of the body" (having the flu, heart disease, cancer, malaria, common cold,...etc) and ill "of the mind" (being a psychopath, narcassist, paranoid, manic, depressive, ...etc). Do the words ill and sick have any predilection towards either of the meanings? And is there a Turkish equivalent?