I didn't interpret their nitpick as saying anything about the spelling of the word in general – also "British spelling" is the usual term, I've never heard anyone call it "English spelling" before. In Australia, we use British spelling for most things, including the word "labour" in most contexts, but the party name is intentionally spelled the American way, due to US influence in its early history. The initial capital is a sign they were talking about the party name as opposed to the common noun