You are DEAD wrong; in applied math, you often need a great deal of creativity to determine what mathematical techniques to use or develop to fit the task at hand...and you DO need to prove that what you are doing is mathematically sound; in fact, the job of a certain type of applied mathematician (known as a mathematical physicist) is to determine whether the sleight-of-hand that theoretical physicists are fond of using is mathematically kosher, and a few times, a mathematical physicist can point out a fatal flaw in a mathematical model used by a theoretical physicist.

There are also, of course, people who do research in applied mathematics itself, and these are most properly known as applied mathematicians.

Your belief that "none of the applied stuff" is used in pure mathematics is only partially true; some of the theorems that have come to be vital to areas like differential geometry, like the local existence and uniqueness theorems for systems of ordinary differential equations, are considered to be part of "applied mathematics" even though they themselves are not all that "applied" in nature...also, much of the study of functional analysis deals with its application to partial differential equations and integral equations. As yet another example, graph theory has numerous connections with algebra and topology and computer science.