Try to use the LineOnLineOverlayer. After, with a tester, you can filter the Feature Class you want to keep and your lines will be split.
Is your base linear feature class otherwise topologically correct? Or are there intersecting lines within it that should not be split (ex. overpass on road)? If the former then either a LineOnLineOverlayer as @philippeb suggests or an Intersector can be used.
If you need to keep your original lines intact, except at the demarcated places, then you need a way to isolate your segments. I suggest assigning a unique ID to your base feature class and then using a spatialRelator with the split class as the requestor and the base class as the supplier with the predicate as intersects. Follow this by a listExploder (if necessary) and then send everything to the Intersector with the Group By set to the unique ID.