Wow! The switching of the polarity seems to have done it! Yay!
I read up on diodes and bridge rectifiers at Wikipedia (
and other places online, and now I think I get it. The bridge rectifiers allow items to work with either polarity, so in my case, it indeed made the short polarity-sensitive. With the polarity reversed, the shorting diode is no longer part of the path used, and thereby the modem causes no trouble. This is indeed the reason there was a difference between the two jacks; the other jack must have already had the opposite polarity.
That everything is now functional doesn't change the fact that the modem is still a little broken, still has the short. The modem has a problem, but we luckily have a workaround. The reason the trouble manifested suddenly on Monday was the modem shorting, something new, and the workaround involved reversing the polarity of the jack, something static. If one of the diodes in the current path being used, the path being used with the current polarity, shorts out, then the modem will cause trouble with a jack at either polarity. (I wonder whether the modem is on its last legs.)
So the problem was in the modem, but we took advantage of the bridge rectifier, and now with the switched polarity that defective diode is bypassed, no longer involved.
What's still weird, however, is the type of trouble the modem's been causing. It isn't that it hasn't been working at all, or that it causes the trouble I've been experiencing only after using it or some equipment daisy-chained through it. It puzzles me that the modem alone, hooked up to the jack pre-polarity reversal, not being used, not turned on, not attached to anything else, only the modem attached to the jack with a good wire, could cause trouble on the line even when the phone I was using was on another jack, when the situation didn't involve a call going through the modem at all. Somehow the defective equipment disturbed the line just by being plugged into the jack, even when it didn't seem to be involved all in the use, not part of what I would think of as part the circuit and in a position to disrupt.
In any event, I am pleased. I am very thankful that I found this forum and to the two posters who patiently and kindly helped me.
Thank you so much!