Whatever "original claim"? Concentration camp was always the official name. Extermination camp was the unofficial name, the whole operation was secret. The official purpose was these were just transit camps in the "resettlement to the East". They were also told they were being sent to work.

The reason why Poland, besides the place that had most Jews in the world so logistic side was obvious, was that Poland (like I just mentioned in https://kotakuinaction2.win/p/12jJUaaUH1/x/c/4JAfAZNrtzc) just didn't exist at all anymore, and so it was a "black hole" where neither German nor any other laws applied.

(Actually the Americans later used a similar logic to set up the X-Ray Camp not anywhere on the American soil but at Guantanamo Bay in a legal loophole.)

In the USSR they needed no camps to rid of the Jews because they just either shot them or instigated the locals to pogroms. But they took many of the Soviet POWs to Poland in order to to kill them, including at the newly build Birkenau Camp that later became Auschwitz II (http://auschwitz.org/en/history/categories-of-prisoners/soviet-pows/).