I think you are mixing up what you hoped would be the relationship between you, your mother and your children with what is reasonable to expect from your dp's mother.
It is awful that your own mother has been a disappointment as a grandma and I can understand that you must feel it watching your dsc having a much better relationship with their dgm.
Just as you love your dc more than your dsc it is natural that she loves her dgc more than her dsgc. Since the parents broke up, as the dad's mother, she may get less time than she used to with her dgc and sadly was a bit thoughtless in focusing on them with her goodbyes in your house.
I agree that ideally that shouldn't happen, but you say she is a lovely person, so I would just keep moving gradually towards a more blended approach. 'Blending' of families, especially extended members, doesn't happen quickly however much we would like it to.
In your position I would be working towards a closer personal relationship between you and your dp's mother. Maybe a lunch or coffee out together or invite her around for coffee when kids are all at school and dp is out. As she grows to love you more and you talk about your life and sometimes your kids more, she will get to know you all better
I wouldn't make an issue over this at first. Have a name your children can call her, preferably not the same as her own grandchildren, there is a difference, then when she's about to leave, if it happens again, after she's said goodbye to her own dgc, say to them, with a smile on your face, say goodbye to nanny/auntie Annie johnny and jane and she will soon get used to saying goodbye to them as well. Check with her first, without the children around, regarding what she wants to be called.
If you take it slowly, recognising the natural differences in the relationship, she could well become a much loved extra family member for your children and a great friend to you. Good luck op 