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Correct
Everything you say appears 100% correct. The current royal family are from the house of Windsor, though this is a name they adopted due to anti-german sentiment in Britain. The house had previously been a German house named "Saxe Coburg Gotha".
The House of Windsor is the reigning royal house of the United Kingdom and the other Commonwealth realms. In 1901, the House of Saxe-Coburg and Gotha (a branch of the House of Wettin) succeeded the House of Hanover to the British monarchy with the accession of King Edward VII, son of Queen Victoria and Prince Albert of Saxe-Coburg and Gotha. In 1917, the name of the royal house was changed from the German Saxe-Coburg and Gotha to the English Windsor because of anti-German sentiment in the United Kingdom during World War I.1
Britannica encyclopedia
This version of history is also shared by the Britannica encyclopedia
House of Windsor, Britannica encyclopedia
The dynastic name Saxe-Coburg-Gotha (German: Sachsen-Coburg-Gotha, or Sachsen-Coburg und Gotha) was that of Victoriaβs German-born husband, Albert, prince consort of Great Britain and Ireland. Their eldest son was Edward VII. During the anti-German atmosphere of World War I, George V declared by royal proclamation (July 17, 1917) that all descendants of Queen Victoria in the male line who were also British subjects would adopt the surname Windsor.
Are the Windsors English? Yes, ever since Elizabeth II (Elizabeth Alexandra Mary) was born 21 April 1926) the Windsors are now all born in England and not Germany, and are therefore 100% English citizens.