"Subject, verb, direct object, object complement" versus "subject, verb, indirect object, direct object"

Reading English Grammar (HarperCollins College Outline, published by HarperResource, an imprint of HarperCollinsPublishers) I found a chapter (Sentence Basics) that explains that in English there are different sentence patterns; in particular

• Subject, verb, direct object, object complement
• Subject, verb, indirect object, direct object

The chapter reports the following example for the patterns:

Alice called Charles Professor Miller.
Alice gave Charles the rabbit.

What is the difference between the two kind of sentences?
What is the difference between the object complement in the first case, and the indirect object in the second case?

PS: The question, which could appear silly, comes from the fact in my first language there is only the object complement.

In the first, "Professor Miller" is modifying "Charles" and is not being acted upon - it's an adjective, and a complement to the direct object, Charles, who is being called Professor Miller.

In the second, "the rabbit" is the direct object - it's being acted upon, that is to say, it's being given. Given to whom? Why to Charles, the indirect object of the sentence.