Ok, thank you for sharing. Now, Cockatoo's are messy birds that throw off a lot of dander and dust. They are very clingy needy birds that are NOISY. Their screech or scream or squawk can pierce your ears. They are some of the loudest in the bird family and from what I have seen louder at times than a Macaw.
African Greys are a bit more quieter, but can be a bit more timid, some can be cuddlers, but some may not be. They aren't as needy as Cockatoo's, but they too throw off dander and dust, to the point some folks can't take it. They are very smart birds and pick up thing quite easily and can talk up a storm or vocalize even during the night time hours.
So either species would require a room all to it's self for sleeping purposes. Birds of this size can be quite expensive to own, it could run a few hundred dollars a month just in food, toys and supplements. Also, a minimum of $200.00 to $500.00 should be kept aside for those emergency vet visits because once a bird starts to show signs of illness, it's almost too late to help them. So it's a rushed trip to a vet or animal hospital.
Do you have an Avian vet near you?
African Grey's as well as Cockatoo's tend to be a one person bird, but this may be overcome with lots and lots of socialization with all family members. But you also have to look at if this is going to be your bird and you'll take it with you when you leave home, the bird may bond with someone else and not YOU.
But out of just the list that you provided, I would go with an African Grey as long as you have a relatively quiet house hold.
But since your family has no other bird experience than Cockatiels and Budgies, I would recommend the following birds as starter birds:
Conures like a Green Cheek Conure
Senegal Parrot, which are supposed to be a quieter bird
Myers parrot
Caique parrot
These birds may not be as exciting as a Cockatoo or African Grey, but if they bite, it will only be a pinch or possibly break the skin and bleed a little. If an African Grey or Cockatoo bites, they can not only break the skin, they can break a bone or bite off a little finger. ALL BIRDS BITE, no matter how tame you think they are, they can have an off day, or your not their favorite person pal that particular day or they get frightened when your interacting with them.
My point here is, this needs to be a family decision since your still living at home and when people buy or home a parrot, I hope they take into account that most live any where from 20-70 years and it is a life time commitment. So if you just happen to get that bird from H E double hockey sticks, you will have to deal with it or god forbid re-home the poor animal.
So I hope you show this post to the rest of your family before making a lifetime and life changing commitment. So everyone involved goes in with a better understanding of what their getting into.