I am pleased to announce that Touch Tone Tommy's suggestion seems to have solved my problem (and if if this success persists, saved a few trees). I connected the cord from machine #1 (the new machine) into the "EXT" port of the old machine -- instead of into the splitter at the wall. (The old machine has a "TEL" port to which the machine's own handset is connected, and the "EXT" port that was available.)
I then sent a test fax to a friend of mine, it went through, and no superfluous paper copy came out of the old machine. Thank you very much.
My 3-line office phone system is a Norstar system, installed around 1990. I hope it continues to function for a few more years. The document I have says M7208.
Thanks to all.