Thanks, Tony.

I thought I'd follow up, because I FIXED IT!

I used a tool that looks like a dental pick (though I'm fairly sure it wasn't, being in our electronics lab at work), and carefully inserted it between the metal "tongue" visible in the jack and the inside side of the jack, then twisted. In effect, this lifted the tongue a bit, as well as pushed it away from the side of the jack.

The insertion point was near the highest point of the tongue, though I tried to work the tongue all along its length.

I made sure to do this on both sides of the tongue, as the left side seemed quite a bit closer to the side of the jack than the right.

After a few minutes of this "Fun With Levers!" exercise, the player was correctly distinguishing between home (AC) and car (no AC).

It seems that the failure mode, at least for my jack, was a stuck tongue. It didn't seem to be broken or to have lost its "springiness", so perhaps there's a bit of a design flaw in these parts that allows the tongue to get stuck in the "down" (i.e., AC plug inserted) position?

That, or repeated use of the plug causes the tongue to deviate enough from its normal non-insertion position that it's always in contact with the plate beneath it?

Either way, this was a dead-simple fix that didn't require opening the unit, and runs very little risk of breaking anything (unless you're hideously strong or have a tool that provides you with excellent leverage, you won't even see any visible give in the tongue while prying it up.)

Hope this helps someone else.