Oh, there is no question that they know. We got little Amy as a re-home when she was 4 months old. Even though she loved us and loved being in our home, it was a full 4 months before she realized that she was staying with us permanently. We could see it in the new way she looked at us.
In your case, Bella probably "sensed" the change in your attitude when you decided to put the adoption papers in.