3-way light circuit controls another room, how can I disconnect it?

I have spent hours troubleshooting this wiring problem we have in our home and can’t find anything wrong, but I am sure I am missing something.

We recently discovered that the 3-way light switch for our basement lights is somehow connected to the wall receptacles and ceiling light in the first floor dining room. Shut off either 3-way basement light switch and it shuts off the power and lights for the dining room. We only recently discovered this glitch, but think it’s been a problem since before we bought the house a few years ago and we never noticed it because 1) we almost always have the basement lights on and 2) we never use the room at night and only have ever used AC in there periodically for a laptop, which has a working battery.

The feed power for the basement light service comes in at the upper basement light switch. From there it uses a 3-way wire going to a junction box in the basement where it connects to the lower basement 3-way wire and the receiving loop for the basement lights. The lower basement light switch has only the 3-way wire incoming; I could identify the wire all the way through and could see no intercepts on the path.

I have traced most of the upper 3-way switch wire until it disappears into the ceiling directly beneath the upper stairway 3-way switch. I have no reason to believe it’s intercepted in the wall, but I suppose it could be.

If I disconnect either side (black or white) of the basement light loop in the junction box, only those lights go out while the dining room stays on (when switched on via either basement light switch).

A strange thing happens when I bypass half of the lower 3-way switch circuit by connecting the black wire from the upper 3-way to the black wire of the basement light loop, but keep the red wire connected through to the lower 3-way switch: the basement lights go on in the up position, but in the down position, the dining room circuit comes on! This is with no black or white wire connected to the lower 3-way switch circuit. If I disconnect the red wires in the junction box, completely disconnecting the lower 3-way wire and switch, the dining circuit then turns off.

How can this be?

If I disconnect the white (rail) passthrough at the upper switch, the basement lights go off, but the dining circuit stays on. This is how the entire circuit should behave when switched. But I don’t understand why disconnecting white wire from the main feed for the basement circuit still allows the dining circuit to stay on.

I decided something is up with the upper 3-way wire that seems to be coming directly from the upper 3-way switch, so I disconnected all leads in the junction box and took a live power reading with the upper switch in the off position:
red – white – 120 VAC,
red – black – 35 VAC,
black – white – 78 VAC

I don’t understand why I am seeing any voltage on the red-black or black-white loops, esp. when one of them should just be off/open. I am completely baffled how an upstairs circuit could be interrupted without intercepting the basement light circuit somehow.

FTR, I have inspected wiring in four of the the six most likely dining room wall receptacles to see if I can find an obvious merge or intercept, but didn’t find any. Using an AC circuit tester on all the upstairs outlets shows proper wiring, no ground shorts or anything. All the wiring here is very tidy, for the most part.

If anyone knows why this might be happening, or other things to test to come to a solution, I would love to figure this out and get the dining room circuit off of our basement light circuit!

Upstairs 3-way, where power feed comes in:

junction box where 3-way lines are merged with basement light loop:

no point in posting downstairs switch, it’s just a single 3-way red, white, black wire connected to the 3 poles on the switch.

Updated also to state I traced the basement lights loop: it’s singular; there are no taps off of it.