After finally getting around to investigating the problem, I easily found a solution in the forums: Apparently, I have to go back to using the rich-text editor, and manually toggle it to use a “source view”.
The underlying problem is likely that Opera (my browser) was previously not supported at all in the editor, which made copy-and-paste of HTML code work per default; however, with some later changes, a partial support was added—and the pasted HTML was now misinterpreted. Toggling the source mode puts the editor back to function as it should.