It could be as simple as Rumpelstiltskin tricking her into getting it back. I just think it’s unlikely right now that they’ll follow the Miller’s Daughter story exactly by having Cora get out of giving up her baby to Rumpel by guessing his name. I know Henry Sr. doesn’t refer to him by name, and it’s certainly possible that he kept his name from people until Cora found out what it was, and then after that he just decides to tell everyone his name.
With regards to Henry Sr, I think it makes total sense that Cora never told him about Rumple. If she married Prince Henry based on the fact that she could spin straw into gold, you don’t tell your new husband that it was all a trick and that you made a deal with the Dark One. He could banish you, annul the marriage, ect. She deceived royalty, so I think we need to bear that in mind. (Just like Cinderella didn’t tell Thomas about her deal with Rumple)