"So I'm still confused. Â In my daughters case, as a SafeHaven baby, is her mother someone you would consider worthy of the title of "birthparent"? Â Or have you read these posts, realized you could never accurately make that judgment and change your mind about who is or isn't worthy?"
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In your daughter's case, I think that's up to you. Not because she's a SafeHaven baby, but because she's YOUR baby and anything she knows about her family of origin is going to come from you, and you need to do what you think is best. If I were in your shoes, I'd probably not throw any other "mother" language into it, because your DD has never been mothered by anybody but you, but I can see the argument both ways.
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"And referring to someone's parent by their first names is one thing (and I would argue that it could be hurtful not to refer to them by the same title that the child refers to them), but eventually, someone's going to overhear you talking and ask who "X" is..."
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If my son wants to talk about his biological mother, he will. I expect that he will talk about her quite a lot, and I HOPE he does because if he doesn't we will never know a blessed thing about her other than the DSS file litany of sins. If somebody asks him to clarify, he will. Nobody will EVER overhear me talking about his biological family, because it's not. my. story. It's not for me to share or speculate with the wider world beyond the bare-minimum version required if there's an ongoing safety or or behavioral issue that can only be handled in context. In which case, I'd probably say "previous caregivers," as in "DS is afraid of men with beards because of a bad experience with previous caregivers, we need to slowly acclimate him to having Mr. X as a gym teacher" or "DS didn't have well-child visits that I know of with his previous caregivers, so we need to do catch-up vaxes for school."Â
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I agree that you are in a very, very different situation because your DD was adopted in infancy, and you have the choice between inserting yourself into the narrative and having there be no narrative at all. I don't need to make that choice.Â
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"will you avoid that title altogether and just say "the person that carried dc in her belly"?"
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I hereby solemnly swear never to let that candy-coated hogwash pass my lips.Â
 For a toddler adopted in infancy? Sure. The fluffy-bunny speak is developmentally appropriate. For a 6 y.o. coming out of the foster system? He'd probably kick me in the shins, and I'd probably deserve it.Â
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I don't have any illusions about having control over what has happened to my DS or what we're all going to go through while he's dealing with it, but one thing I DO have control over is my own mouth, and the shutting thereof, and the value-neutral terms that come out of it. First names are good, for me to use.Â