My kid is atypical. A little bit intellectually disabled, a little bit schizoid, very ticcy, and entirely dependent on me. But here's how it was for us - I told her when she was 11, accidentally blurting it out in frustration after she cracked up laughing when I suggested I might try on her nail polish. I should have told her years before that, but I thought she knew. My outward transition by then was happening gradually but was very obvious.
Her initial reaction was sobbing. She had thought I was a gay woman (we had been involved in lgbt and civil rights stuff all her life) and she had her heart set on my marrying another woman so she could get a mom who wore dresses and makeup. I never knew before that she thought that, or wanted that, and so my years of trying to be reasonable and acceptable only set her up to be confused and disappointed. I regret waiting so long.
I tried to comfort her at the time, but all I could give was hugs and reassurances I wasn't going anywhere. Her tears subsided almost instantly when she realized that she could have a DAD, omg a dad. She thought I would instantly become a hulking manly man and all danger would be null and void. She rambled half that night about how powerful I was and what problems I could solve. I thought I had raised her to be less rigid in her gender definitions, but I can't control society. We had only recently left alabama.
I told her she could call me mom as long as she needed to, but she quickly switched to 'dad' of her own volition. She would use mom occasionally, and the only pattern I could see was it seemed if someone knew I was trans she would use mom and if nobody knew she would call me dad. When we were alone it was dad all the way.
She was sometimes embarrassed of me, but that depended on who was around. We have since stopped hanging out in places where gender or sexuality is overpoliced. Transpeople on the radio these last few years seem to have helped her see my "problem" with gender as a real and global phenomenon. She is much more accepting and willing to treat my being trans as more than a weird secret when other people have the same problem, and the problem is framed as normal people facing barriers to being themselves.
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All that is a whole lot of words about my own child (who is different), when what I mean is your kid will accept and respect you if nobody important to your kid doesn't. Your children will follow the lead of the coolest kids in their classes and the most-liked family they know. I do agree with what's been said about kids these days being more aware of the variety of human experience than most of us were growing up, and that it's not fair (to other transpeople's existence or to cispeoples' intelligence) to treat gender identity as shameful or unimportant.