IMHO, I believe that the Simple CASE and the Searched CASE are both misnamed. In fact, the names should be interchanged. In other words, the Simple CASE should be called the Searched CASE, and the Searched CASE should be called the Simple CASE.
The Simple CASE should be called the Searched CASE because the input_expression is effectively being searched among the number of when_expressions for equality.
The Searched CASE should be called the Simple CASE because its syntax is much simpler. Both contain the following syntactical constructs: (a) the CASE key word, (b) one or more WHEN/THEN clauses, (c) optional ELSE clause and (d) the END key word. The Searched CASE, however, contains an additional syntactical construct: the input_expression.
Perhaps we might get it right this time?