Insertions in Red-Black Trees
Yes, a given set can be represented by multiple red-black trees, and this works incrementally, at least some of the time.  That is, there exists more than one valid red-black tree insertion algorithm, whose outputs are not always equal.

The easiest way of seeing this is to focus on the correspondence between red-black trees and $(2, 4)$-B-trees.  A black node with its red children correspond to a B-tree node.

In particular, if you have two sibling B-tree nodes with four elements in total, they can be distributed 1/3 or 2/2.  In red-black terms, if two black nodes are cousins, it's possible for each to have one red child, or for one of them to have two red children and the other to have none.  More than that, you can shuffle the elements around between the nodes locally to transform one such pattern into the other.

As wikipedia says, "The balancing of the tree is not perfect, but it is good enough to allow it to guarantee searching in O(log n) time"; the imperfection of balancing gives room for multiple tree representations of the same set.