Difference between intended interpretation and extended interpretation in first-order logic
You should only care about intended interpretation.  I don't like the concept of the extended interpretation.  The latter is only included to make precise the concept of quantification, but I think it makes things more confusing instead.

Before understanding what an intended interpretation is, you should really know what a model is, and what a language or signature is.  To follow their example:  A language can be consisting of the constants Richard and John, the predicates (the arity in backets) Brother (2), OnHead (2), Person (1), King (1) and Crown (1).  Finally you have the unary function LeftLeg (1).

That was the language, and though we use words as Brother, this should really impose no meaning, we could just as well have written it A.

A model, on the other hand, or what they call an intended interpretation is a domain, together with *interpretations of the above predicates, constants and functions.  Here's an example:


Domain = { Richard, John, Crown, Dog }
Brother = { (Richard, Crown), (Dog, John) }
OnHead = { (Dog, John), (Richard, Dog), (Crown, Richard) } (the crown' on Richard's head, Richard's on Dog's head, Dog's on John's head)
Person = {John, Dog}
Richard = Richard (oops, heavy overloading!)
John = Dog (oops, heavy overloading!)


I didn't complete the definition, but I hope you get the point.  If you put all those above together as a KB, (Domain, Richard, John, Brother, OnHead, Person, etc...), you get your inteded interpretation.

The extended interpretation you should ignore.  Completely.  At least if you understand what $\forall x P(x)$ means for some first order sentence $P$.  And $\exists x P(x)$.