Those are the ones he's referring to. They're called waribashi, and his advice against rubbing them together is entirely correct. They may be disposable, but the act of rubbing them together implies exactly what he notes, and is considered poor manners at a Japanese table.
If you still want to check for splinters or loose shavings, this can be done more politely with a discreet rub/roll action of each one between a finger and thumb across (not along) the edge where the cut between them was made.
Bonus: If you really want to do things right, after you're finished eating, slide the chopsticks back into the paper wrapper. This is a consideration made toward the person who has to clear the table, as it makes things easier, and a bit more sanitary, for them to clean up.