This time, in Qufu, Shandong Province, the fellow on duty decided that neither a foreign ID card, nor a passport with a current Chinese visa would do. After rummaging in my wallet, I decided to make yet another try with yet another document: a reader's card from the National Library in Beijing. (Not that I ever *read* anything there, but you needed to get a card to get on premises, and getting it was free...). No name or photo on the card, but it was officially looking enough... and bingo, it was accepted!
P.S. As I've looked at more of such places (in Jiangsu mostly), it seemed that the usual mode of operation for them was to swipe a PRC national ID card, which, I assume, all resident citizens are supposed to possess. This way, at least potentially, who uses what becomes automatically loggable (since the login ID given out by the internet cafe may be linked to one's card number). I am not sure what exactly happened when a particular place accepted my passport, a foreign ID card, or even a library card (well, just that once) as an ID; it appeared that in some cases they simply ended up swiping someone else's national ID card on my behalf.
(Poster above: "Present an ID and register with your real name! Civilized behavior on the Internet, and healthy leisure!". Seen in Gansu, 2009).