Babka - it is unpolite word - if someone calls his grandmother babka it means he does not like her.
Babcia it the proper one, babusia, babunia, babus are made for the very loving babcia by their gradnchildren. And from these words you get words that show even more affection - busia, bunia etc.
Baba - it is toddlers word for babcia - as you can see it is much easier to pronounce.
Babushka - we used to call old women in scarfs babushka because it is russian word for babcia and all older women in Russia used to wear scarfs - there is no such word in Polish - this is russian word!
Dziadek is the grandfather - dziadzius, dziadzia if you want to show your love and you are more bounded to your grandfather.
It is the same with mama - mamusia - like in English mom - mommy. In English you have grandmother, granny, nanny, nana - you probably would call your grandmother "grandmother" but granny i.e and it is the same in polish - I call my grandmother babcia, my kids used to call their maternal grandmother babus and paternal babcia - the diffrence in names is because they are more bounded with my (maternal granny) mother than my mother-in-law. And it does not matter from which part of the country you are - it depends more on the family and in polish immigrants case - how well they speak Polish :)
Dziadek is the grandfather - dziadzius, dziadzia if you want to show your love and you are more bounded to your grandfather.
and i forgot - dziadzia is the toddler word for dziadzius as it is easier to pronounce for babies - it is different with immigrants as they always choose the easier form hence your dziadzia, baba, busia (shorter is easier than original babusia) or "babci"
- oh I know why you call your babcia - "babci" - probably because you could hear it always - "go to your babci" - idz do babci - this is polish grammar - who? - babcia , to whom? babci :)
I hope you could understand my English ;)