The use of ‘the continent’ here seems to clearly be a colloquial rather than academic use. Notice the second term given is ‘Mainland Europe’ implying there is some area of Europe that is not ‘Mainland’. Yes there is! The European islands.
In fact, what you are citing here actually proves my point. It states, and I quote, “the continent of Europe, explicitly excluding European islands”. That is pretty clear, that the terms it mentions are talking about Europe (that is everything), but excluding the islands that the text clearly acknowledges are part of Europe. The very reason these terms exist is to narrow down the European continent in conversation to ‘Mainland Europe’ (that is, excluding islands).
Just to further illustrate this, I understand your skepticism of Wikipedia, but a simple Google Images search of ‘continents of the world’ will bring up a library of maps from a multitude of websites. You’ll notice every single one includes Britain and Ireland in Europe and Japan in Asia.
