You see people care about this stuff because they go to their therapist and say, "boy there are a lot of neurotypical people I have to predict, and being slightly better at reading autistic people doesn't make up for it."
In my experience people are often bad at explaining their inner emotions. Rationalisation is one of my favorite words!
I will often ask a friend why they acted a certain way, and I will carefully listen to what they say, but I often disagree with their stated motives - denial and lack of self-awareness are rife. This has nothing to do with people consciously trying to deceive me (there's another topic!). I have the same faults. I'm average at reading people, yet all too often I have to trust my own flawed instincts about what I think is going on.
The second definition makes more sense to me since it also comes up in non-autism diagnoses, like personality disorders. They have trouble reliably predicting what other minds would think, but for different reasons.