Using ArcGIS 10.3, I would like to cut the river lines at the edge of water body polygons bordering the region. Using the intersect tool and split line to points tool resulted with the disappearance of the lines located outside the water body polygons. Any idea how to split these lines at the border without erasing them?
According to this ESRI page, you're doing the right thing using Intersect then Split Line at Point. Did you check the Intersects tool's Output Type field? If it's set to Input or Line, you will lose the lines outside the border. Set it to Point before using it in Split Line at Point.
Split Line at Point method
Use the Analysis Tools > Overlay > Intersect tool to find the points where the polygon boundaries intersect the lines to generate a feature class of type Point. That is, the tool’s Output Type field is set to Point.
Use the output points in Data Management Tools > Features > Split Line at Point tool to break apart the lines at the point locations (this tool requires an Advanced license).