To split area’s in iD-editor I copy all tags from the area and save them. Then you can select a node of the area where you want to split it an press X to cut it in halfes. You will get a multipolygon relation. Now delete all tags from the area, and delete the relation (remove it from the area). That result in 2 independent ways. These now should be drawn around where you want the new area’s and then closed to get an area.
You can select the last node and add more nodes with “A”, like when you draw roads to the end.
I prefer to use the one with the history in the bigger area. You put the saved tags on both of the area’s, and have two of them without drawing all new.
Depending on where iD set the opposite point to split, you have to redraw more or less of the area, but in most cases its faster then cut area and draw one completely new.