When she asked the Doctor if she would share what Sally said, and the Doctor answered No, that that was between her and Sally and that anything Betty said to the Doctor was Confidential as well. And Betty "That's better". So Betty liked that the doctor would not betray Sally the way the Doctor who Don found betrayed Betty.
Oh, see, I totally read that the opposite. The "That's better," rang with a slight bit of disappointment, akin to "Oh, yes, of course," as if it's what would be expected, but underscored with disappointment because it's not what she wanted to hear. Betty comes from a background where secrets weren't allowed. And she was monumentally betrayed by secrets being kept from her. So she's torn in half on that score. She wishes she'd been able to keep secrets (and has in fact, kept secrets herself, with respect to her backroom tryst at the bar after she found out she was pregnant with Gene and of course, her burgeoning relationship with Henry), but she was never able to keep secrets from her mother at Sally's age, so why should Sally be granted a freedom she never experienced?
Or I could have it completely wrong.