Is any computational problem a search problem?
The linked PDF, "Handout 2
 for Stanford University — CS254: Computational Complexity", defines four types of "computational problems":


Decision problems
Search problems
Optimization problems
Counting problems


As the document notes at the beginning, it focuses on the first two: decision and search problems.


  In this course we will deal with four types of computational problems: decision problems, search problems, optimization problems, and counting problems. For the moment, we will discuss decision and search problem.


However, the document seems to talk about decision problems and search problems as though they were conceptually distinct from each other.  It also fails to mention the mapping between them, which is simple and commonly known (PlanetMath):


  A relation $R$ can be viewed as a search problem, and a Turing machine which calculates $R$ is also said to solve it. Every search problem has a corresponding decision problem, namely $L(R)=\{x\mid\exists yR(x,y)\}$.


In short, you're correct to realize that decision and search problems are two sides of the same coin.  You're also correct that it's weird that the document doesn't appear to mention this obvious fact.

Personally, I'd tend to see these terms differing in connotation.  This is, while they can often be mapped to each other, referring to a particular problem as one type rather than another suggests a perspective on how to see that problem, rather than asserts that that problem can't be cast to another type.