What I think you have done is to select the node on the junction, and then delete it.
You actually want to split the road in two and delete one of the two parts of the road, so undo that change, without saving and select the last node on South Signal, before the junction, use the scissors to split it there, and then select the road between it and the junction, and delete that.
As it seems unlikely that that last node is the real end of South Signal, you may have to then drag it into a better position.
The procedure I gave above adds an extra node to mark the true end of South Signal, first.
If you add a line to fill in the gap you have made, that line will have no tags, and will not be considered to be part of the road. Fixing that is going to be more difficult than splitting South Signal and deleting part of it.
(Whilst you could drag one end of the gap onto the other until it snaps, that will change the position of the boundary between Buckboard Lane and South Montgomery, and also change the shape.)
I’ve just tried another way, which is actually potentially better:
Select the node on the junction.
Use the icon just below the scissors (the one with two blue arrows) to split the node into three.
Drag the node. If it moves the wrong road, remember where it should be and try dragging again.
If you get the correct road, drag it back to where the road really ends.
Finally drag any other road endings that you moved back to their original position; they should snap together.
Check that they are rejoined by trying to move the junction, and then undo that move.
If they are not joined, moving them back should make them snap back together. (When I dry ran this, South Montgomery moved first, then South SIgnal, so I just had to move South Montgomery back to join Buckboard. The reason I don’t actually make the change is that I want someone who has actually seen what is there to be recorded as the person making the change.).
(The advantage of this method is that there is a danger, when splitting a road, that the split will be achieved by creating a new section for the part you want to keep. That means that you lose the history information. The disadvantage is that you risk moving the junction between the roads you want to keep, because iD doesn’t allow you to detach just one road from a three way junction, and doesn’t seem to allow you to select which node to move.)
I have added a map note http://www.openstreetmap.org/note/843317 which is the slow way of getting the change done, as it relies on some, preferably local mapper, seeing the note, and verifying the true situation. There is a note close to this location that has been there for over two years. Another approach is to contact the person that mapped the road, but that happened 7 years ago, and the person doesn’t seem to have been active for 5 months, and may no longer be in the area, if they ever were.