I am pretty sure I know the answer, but better ask and seem stupid than do something wrong and be stupid
If I split a way, the two resulting ways remain joined? In other words, I am un-merging, not un-joining? I want to split a way at the intersection of several ways (click here) to make the section of the yellow way east of the junction one way.
When you split two ways in either Potlatch (‘x’ or cut tool) or JOSM ('p) the two child ways remain joined at the node where they were cut. In Potlatch this will be clear by the outer square box round the node.
To remove the join (i.e., create two nodes instead of one) you can use Shift-J in Potlatch or “g” in JOSM. “j” works in both tools to re-glue the nodes together.