You are right in that the book is mistaking. I also get a LaTeX compilation error, so hopefully this is understandable: As you seem to have proved, it holds that the set
ext( cap K_i )
and not the other way around like the book says.
And you need not even have equality for a finite number of sets. You can take the sets that the wikipage used as a counterexample for closures, i.e.
H = (0,2) cup (3,4), and
K = (1,3).
For these sets, I get
ext( H cap K ) = (-infty,1) cup (2,infty), and
ext(H) cup ext(K) = (-infty,1) cup (2,3) cup (3,infty),
so compared to the first set, the latter set lacks the number 3.