I also have a similar situation in my family. My "dad" (who is technically my ex-step-father, but raised me), lived with a woman for many years after his and my momís divorce, and her daughter (a few years younger than me) lived with them. So, when my younger half-sisters and I went for our every-other-weekend visitations, there were technically four daughters in the house.
I think largely because of "oldest child syndrome" so to speak, she and I constantly butted heads. My younger sisters just had a second older sister who they sometimes got along with, sometimes didnít. As a result, they typically consider her their step-sister, and call her that on Facebook (or just sister) while I consider her more "the daughter of my dadís ex." She and my dad also still have a relationship.
Weíre friends on Facebook, and I sent her a homemade quilt when she had a baby (mostly because I wanted an excuse to make one), and we are friendly when we end up at the same events (rare, since I live across the country), but we donít talk otherwise. She has made references to me as her step-sister though.
Related, I call my bio-dadís wifeís daughters my sisters (or step-sisters when clarifying), not necessarily for the legality of our parentsí relationship, but because we do have a relationship. I visit them when I go home. I consider their kids my nieces and nephews, and hope to have a life-long relationship with them. I also have a half-sister on that side, whom I claim as sister because of the blood relationship, and because of my nieces and nephews, but she and I donít have a close relationship. My step-sisters do not usually call her a sister, because while they tried to maintain a relationship with her, there never has been one unless half-sister needs something from them. No amount of parental intervention is going to change that, and one step-sis did have to tell her Mom "look, just because youíre married to her dad, doesnít make her my sister. A relationship and effort on her part would, but sheís not willing to do that. So, stop."
Long story short, I can totally understand how sheís not your sister, but you may be hers. And thereís nothing wrong with that difference in levels, unless she is coming to you and saying she feels hurt.
Itís not your mom's place to step in and try to force a relationship, or even pass on Bettyís hurt. To her, I would say, "Mom, I recognize her at the level I feel comfortable with. She has never had a sisterly role in my life. If, someday, we have a closer relationship, I may change my mind, but that is between her and I. Please donít try to guilt me with second-hand information."óor whatever fits your feelings.