I think either Emilia has some relation to Satella or puck definitely has some past with Satella ( ofcourse who am i kidding... I mean puck is her mother).
You might remember in episode one puck asked Emilia reason for introducing herself as Satella and later commented that she really have terrible taste.
As Satella name's connection is a secret to only Emilia and puck so It reveals her liking towards Subaru as she presented herself as the one everyone hated but in reality it is something which is very close to her. Also as Subaru was new to town her consciousness might have mixed up with the things she love when she saw Subaru.
And after resurrection when she got mad at Subaru fit calling her Satella it was normal reaction which she would give to anybody to fit in with people.
And the explanation for Emilia getting angry at Satella name might be that she doesn't get angry on people calling her witch Satella but the expression people make on her while saying that makes her sad.
P.S.- I don't know if I am able to convey this to you guys. And I might be right or totally wrong. A 70-30 chance......