Manchu state was established beyond China, and it is clear by the following reasons.
The map you quoted at http://en.wikipedia.org/wiki/File:Ming-Empire2.jpg belongs only to the Yongle time (1360-1424).
Before and after the land of Manchu was not a part of the Ming: http://depts.washington.edu/chinaciv/1xarming.htm
Refer also to the section "Manchuria during the Ming Dynasty: http://en.wikipedia.org/wiki/Manchuria and to the history of Jianzhou Jurchens: http://en.wikipedia.org/wiki/Jianzhou_Jurchens. Also see Cambridge History of China vols. 8 and 9.
The Han Chinese called the Manchus "guan-wai de ren" ("people from beyond of outposts") meant that the Manchu lived outside China.-SK