The definition for a Prufer ring given by Butts and Smith in "Prufer rings" (1967) is
Definition 1. $R$ is a Prufer ring if and only if, for each $P$ proper prime ideal of $R$, the ideals of $R_P$ are linearly ordered.
Another definition is
Definition 2. A ring is a Prufer ring if all its regular finite ideals are invertible.
Now what I want is to prove the equivalence between these two definitions. Any link to a book or article where this is done is welcomed, as well as any hint.
Your two definition are actually not equivalent, although they are in the domain case. Definition 1 is what is usually challed an arithmetical ring. Check out this note. You can see from Theorem 2.15 that the definitions are not equivalent. Also you find some nice characterizations there.