Deletion from a red/black tree

In Introduction to Algorithms by Thomas Cormen insertion and deletion is said to take O (lg n) time.

I have read that delete takes amortized constant time if you supply the pointer to the element being deleted (you can omit the searching) but that is exactly the version described in cormen and it is said to take O(lg n) time, so how does this fit together?

Another thing, delete also use tree-successor which runs in the height of the tree so this would also slow down delete to O(lg n).
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