My grandmother told me England was not part of the European continent. I got an answer wrong on a test because of that. She refused admit she was wrong even after I showed her in my text book.
It is right there on the Eurasia map at the link you shared, and on the list of Eurasian territories, so OP was correct.
The thing is that "continental Europe" is not the same as the continent of Europe, which does include the islands. Mainland Europe is a less ambiguous name.
The largest Eurasian islands by area are Borneo, Sumatra, Honshu, Great Britain, Sulawesi, Java, Luzon, Iceland, Mindanao, Ireland, Hokkaido, Sakhalin, and Sri Lanka.
Okay, then the "politico-cultural concept" and common usage of the name "Europe" is of it being one of the major continents, regardless of if it's on its own tectonic plate or not. It clearly includes the British Isles.