I don't think that you're weird, or that he's weird, or anything.
I think it's normal to feel that detachment over something so traumatic - there's nothing wrong with you for not crying or feeling much yourself. It's hard to live with that pain, so it's a natural response to block it out.
I also think it's alright that you feel awkward that he cried - I would feel that way too. When someone cries for someone else, it shows that they care and have empathy for that person. I know that I'm not used to feeling like someone cares, so if someone shows that in a concrete way like that... I don't know how to respond.
I also think it's alright that your T cried, it shows that he cares and sometimes people just can't stop the tears.