I mean, it just sounds silly to say "The Lion King is a remake of Hamlet..... with Lions..." Because so much is different, besides just the lions
I'd go by the actual definition:
"a movie or piece of music that has been filmed or recorded again and rereleased."
Being inspired by and/or adopting many elements of a piece of work is not a remake.
There are similar themes in any story, and, if you stretch the comparisons far enough, you'll say one is a remake of the other.
ET came back to life, so is ST 3 is a remake of ET, because Spock came back - and both are a remake of a certain story that took place 2,000 years ago.