I agree with the other moms on here.
I don't think she's being protective of her baby sister. (she loses interest as soon as "dad" walks away) I think that she's just worried that he's going to stop doing all the things he's done for her up until now. She's worried that "now he has his 'own baby', and won't want me anymore." It's just going to take some time and work. Try sitting down and talking with her (like you said you would lol) and ask her how she feels, what she worries about, things like that. I know she's only 9, but I'm sure she understands for the most part. She may or may not tell you guys everything, but just let her know if she has a problem or anything, or just wants to talk, that you guys are there for her. Both of you. That way, she knows you guys well always be there for her..no matter what. You guys also need to talk with her about how the baby needs one on one time too. Let her know that it's not that you guys don't want to spend time with her, it's just that she has to learn how to "take turns" now. There's only 1 mommy and 1 "daddy" (don't know what she calls him) and 2 of them. I also think that you should get her set up with something else when he wants to spend time with the baby or have you take her to do something with you. Here's an idea, (if this works for your husband) take your 9 year old out with you (shopping, going to the mall, whatever) while "dad" stays home with baby. That way he gets one on one time with the baby, and she doesn't feel left out.
Also, do you guys all spend time together as a "family"? Like you, your husband, and your 9y.o. all play with the baby together? One on one time is important, but family time is just as important! lol Oh! and it might also help if just "dad" spent time with both of them at the same time. (if he doesn't already) Have them play games together! Like peek-a-boo with baby or "tickle time". It would be good if "dad" holds the baby and "plays" with the 9y.o. Like one thing we do (I've been a "single" mom of 3 boys, so I've had to do this twice! lol) is I hold the baby's arms up and tell my other boy(s) "Oh! Get the baby! Ahhhh!" and they "tickle" him all over! It's really cute and we all take turns "getting the baby" lol Idk what else to say right now, and this is getting kinda long lol, but I have more ideas if you need more help. Just send me a message if you want to talk! :D
Anyways, I hope this helps! Good luck! Just remember, it's just a phase...it will pass with time and work. She's only 9..this is the only way she knows how to "deal" with it.