I don't understand what point you are trying to make.
Anyway, you are wrong. Within the part I quoted, they are differentiating their support for conservative figures like Reagan or Thatcher, with their championing of liberal values - which they mention such as gay marriage and gun control. They are thus using the modern terminology, nor do they capitalise liberal. Elsewhere in the same article, they reference their Classical 19th century Liberal credentials, where they do. Further, in that quotation they firmly state they consider themselves Radicals - which is per definition, not centrist. 19th century Liberalism was socially on the Left of the spectrum besides, as evidenced by Catholic Emancipation and such.
Lastly, Euthanasia is nowhere a centrist opinion, except maybe the Low countries. Actively arguing for massive changes in the Laws of a nation does not qualify as such, and the Economist does not argue for a moderate form thereof to boot.
The Economist tends to support free trade, which was Classically Liberal, but is today thought conservative (prior to this spate of Protectionism), but its support for broader EU outside the common economic community certainly is not. Socially though, it sits clearly on the left of the spectrum, as that quote above is quite emphatic on (gay marriage, gun control, prison reform, decolonisation, etc). So as I said, I think they are a fairly equitable centre left publication - you may disagree with me; but I see no reason to think otherwise and their own description of their beliefs points to it.
