I have some answers, for one, and also, my AM radio is now working. I purchased a workshop manual, which has the identification and location of all the units in question, and this probably helped solve this:
1) On my truck (2009 LR2 with Tech package and Nav), the AM and FM signal booster is one unit, in both a non-diversity and diversity system*, and is located under the headliner, just above the right rear window, roughly centered to the window. It has two RF connections, and a snap-on connection to the antenna(s) on that window.
2) The unit in the rear hatch, the one to the left, under the cover for changing the high brake light bulb, which I have (and which is white on my car), is the "Twin FM antenna for diversity tuning"*.
3) The unit my car does not have, but which I've seen on other pics (including the one I posted in my OP), and which is on the right, under the cover for changing the high brake light bulb, is/would be the "VICS/TMC antenna amp". This is for a system that some countries have, which you can look up... I won't get into it here. But I don't need it, and it does not relate to AM or FM radio.
Then, what is important to know, is that these AM/FM and FM antenna boosters send that boosted signal to (of all things) the DVD unit for the Nav system, under the front right seat. The path of the AM signal is from the antenna (that snap connector), to the booster (under the headliner), to the DVD unit (under the right seat)... where it is somehow processed, and then sent to the radio unit in the dash.
What I did: Most electronic problems are connections, contacts, bad solder joints, while failed components are in my experience the last problem seen. So I started by taking down the headliner enough to "eyeball' the AM booster. Everything looked connected. Then I reached in, and jiggled the connectors... all of them. They seemed tight. Then I went under the seat, and jiggled and tested all the wires and connectors there... I did not yet identify the ones from the booster, it is a rat's nest, and I put it off for later.
I didn't check the AM radio because I didn't do anything... I thought. But the next day, on a hopeful whim, I tried it: And it worked perfectly! Later, there was some break-up... not static, I don't think, but some break up. This seemed to be more apparent on a rough road... then, it went away.
Conclusion: There is a bad connection or ground somewhere from the AM booster to the DVD unit. Maybe a pinched, or internally broken lead. Maybe a slightly corroded or tarnished connection. Maybe a bad ground, where a unit is not making good contact with ground. In any case, my jiggling and messing around got it going... So I am going to take down the headliner, take out the booster, and use contact cleaner on all contacts. Then I will check the grounds. I will do the same for the DVD unit. But basically, I think the problem is solved.
* Diversity tuning is a system which has more than one antenna source, and then automatically switches to the source with the strongest signal.