Trying to explain that the multiple reasons of the inability to sing the right tone (to match pitch) I encountered two other terms: amusia and ton defness.
I wonder how far these two terms are meaning the same and what is the difference.
Neither are particularly precise terms.
'Tone deaf' describes people who have trouble with recognising pitch. 'Amusia' would also include problems with other elements of music like rhythm or timbre.
Or that's what musicians would say 'tone deaf' means. The wider community might apply it to someone who just doesn't 'get' music in any way - appreciation or ability. Even wider, there's a non-musical meaning - someone who can't read a social situation.