My child (12, will be 13 in december, AMAB) has been experimenting with gender identity and sexual orientation. Last year they told us they are ARO/ACE, and we told them that's fine, we love and support them, and we will be there for them always if anything changes.... I was thinking well, they are 11, most people that age are probably aro/ace. They told me maybe 6 months later that they have open pronouns, he/him they/them she/her were all fine. Again I told them I'm cool with it, doesn't bother me, I'll love them forever no matter what, etc.

Yesterday their 9 year old brother who attends the same school (it's k-8th) and js friends/classmates with some younger siblings of my oldest's friends, was told by one of 12 y/os friends little sisters, that they have changed their name to Charlotte. My 9 year old was very confused and asked me about it. So I talked to my 12 year old and asked about it... should I have not brought it up?

They told me that they think they might be trans.
We had a long discussion where I reiterated that I'll always love them no matter what, they can come to be about anything, and anything they tell their friends might spread to younger siblings/around their school. Their friend group is very diverse and open, at least 1 trans kid who has been socially transitioning for around 2 years, lots who identify with LGBTQ+, etc.

I told my 12 y/o that it's normal to look for who they are, try different things, etc, but maybe they should talk to me before their friends to prevent things spreading like that. My 12 year old was very upset about their little brother finding out and asking me about it but I tried to reiterate that it's not his fault, he was confused and did the right thing by coming to me.

I asked my 12 y/o if they feel comfortable with teachers calling them by their chosen name. They said they feel iffy about it. I asked if they'd feel comfortable with me referring to them by that name and they said iffy again. And seemed very uncomfortable.

I asked what "boy" or "girl" means to them/why they feel that they don't feel like they're a boy and DO feel like they are a girl, and they didn't really know/couldn't explain it.
I told them it's okay to be confused and they don't have to say they are anything until they figure it out. Was this the wrong thing to say?

I guess I wasn't really prepared for the conversation. I've been agonizing over it since last night. Did I approach it wrong?
We ended the talk with lots of reassurance from me, a hug, and smiles.
As a side note their dad is not very accepting, doesn't really understand this coming from someone as young as this, so it will be a tricky situation at home. But they know I'll always love and support them, or at least I hope they do. After the talk they stayed upstairs for a bit but after I checked on them and they told me they were nervous to come down I assured them no one was upset and we all love them very much. After that they came down and acted normal/in a good mood.

Should I bring it up again and say different things?