Like you, I can't tell the difference between casual acquaintances and friendships well, but this situation, I feel, isn't just about that, or necessarily about being a mind reader (though I do think that would help).
It's about not making each situation about you and your feelings, often unconsciously. Most people, even NT people, do this, some more than others, and don't even realise it. OP isn't being selfish deliberately, or misreading cues, she's being human. Humans are inherently self-absorbed, it's a survival trait. Some people (and I am a long way from being one of them) don't have this trait to such a profound level, or have learnt techniques to tame it.
If we look at a very simple exchange example the OP gave us: "I'm there for you". Sounds great, doesn't it? Sounds like a lovely friend. OP certainly meant it with the best intentions.
But that sentence is, fundamentally about the OP. It's telling the other person something about the OP, rather than asking the person about themselves. Bearing in mind that the other person in this situation is not only also human, with that same self-absorption trait, but has an added layer of grief on top of that, subconsciously what they actually read isn't 'I'm there for you' as an offer of help, but 'I'm there for you' as a reminder that OP exists, and on top of everything else, they've got to think about OP. It can read as though OP is asking to be included in something that isn't about her.
So, how could OP reframe that, if she were to find herself in that situation again? Instead of stating that she's there, she could simply ask, 'how are you?'. That puts the other person front and centre. It takes the 'I' out of it, and makes the recipient feel, again, subconsciously, that the OP is thinking about them, rather than wanting the recipient to think about her.
Dealing with others' grief is so very hard for those outside. When I met the man who would become my husband, he was dealing with a terrible close personal tragedy. The biggest lesson I learned through that really intense experience, was to listen.. And asking how someone is, shows you want to listen.
I'm still not good at the art of listening in general. Nor have I got rid of my tendency to think about myself. It's taken some bloody hard honesty to admit that sometimes, for me, 'being there' wasn't something I did for the other person, it was something I did for me. To show what a good friend I was, to boost my value in the eyes of the other person.
I've got a friend who has always been good at this. So good she became a Samaritan. For some rare, beautiful people, it comes easy. For most of us it takes effort. For some of us, it's much harder. And it isn't because those people are awful people. It's just a skill it takes us more effort to acquire. But I do believe we can acquire it. It will never be easy for us, or natural, but being aware that it's something we aren't good at is a big first step.