Thanks for those pics!
Right, I learn that when they went from the 4-speed TS250 to the 5-speed TS250/1, that lip was introduced. I had not noticed that lip on your earlier pics, and just reminding you that I never saw any of those barrels in-the-flesh. However, that lip is also there in the later ETZ's, which followed the 5-speed TS. It probably helps sealing, but I don't think it is essential: when boring a 250 barrel to 300, as some do, that lip is lost. However, it will probably require a different head... Either to accommodate the lip, or to compensate different heights - you should measure the heights & distances as indicated in viewtopic.php?f=20&t=10396&sid=a1e9473259d5d867a7cb99b1e9f93366#p51000. I suspect that the port heights from the top of the lip will be the same as those from the top of the barrel (sleeve) in the 4-speed one.
For clarity, I assume you previously had the 4-speed TS250 barrel, without the lip, and that the one you bought was the 5-speed TS250/1 barrel, with the lip. Please correct me if I have this wrong. Your bike is a '78 TS250/1, = the 5-speed one, so it had the barrel from the previous model on it. And maybe with a head matching that barrel, maybe with the original TS250/1 head (in which case the compression ration will have been way too low!).
On your observations:
1) I think the standard bore & stroke were not changed, and are still the same in the ETZ: 69.00mm & 65.00mm. Now 1.5mm difference in circumference means 1.5/pi difference in diameter, which is 0.5mm. So they are not both at the same oversize, the 250/1 is one size bigger.
2) The lip needs to fit inside the head. At TDC, the piston will probably be at exactly the same height from the top of the lip in the TS250/1, as it is from the top of the TS250 barrel. But you need to verify that. If that is so, you might still use a TS250 head on the TS250/1 barrel, but it requires a ring around the lip, to correct the compression & the height of the barrel. A bit like this one:

- Compression ring.jpg (21.86 KiB) Viewed 8 times
but then with the thickness of the lip's height. But as your bike is a 5-speed one, you might actually already have the correct head? We don't know what your predecessor did, have a look. At any rate, you should measure the gap (page 30 of the manual):

- Gap in TS250.jpg (30.73 KiB) Viewed 8 times
3) Haven't a clue.
On your question: I think you bought the barrel that belongs to your bike, but you previously had an older model barrel on it. Yet to fit it, you may have a problem with your head - see above.