There are several dozen 'classification' theorems of various notions of $p$-divisible groups in terms of crystals (where 'crystal' can seemingly mean various things), thanks to theorems of Dieudonné, Grothendieck, Messing, Mazur, Manin, Bertelot, Ogus, Cartier, Fontaine, A. de Jong, Breuil, Kisin, Scholze, Weinstein, and likely many others. As a newcomer I have no idea how to untangle all this mess. Could someone give me a clear overview on the following?
How do all these different classification theorems relate? Do I understand correctly that they are all just different special cases of an envisaged classification of $p$-divisible groups over an arbitrary base?
What is it that motivates people? Messing showed that $p$-divisible groups over arbitrary base schemes with $p$ nilpotent are in correspondence with certain crystals. Aren't we close to done then?
I am completely OK with skipping technicalities. I just want to get an overview of what is going on.