All cut types other than On Line require that a cut path be a closed path. This is because there is no self-evident 'inside' or 'outside' of an unclosed path. When presented with an open path, Origin will default to an On Line cut type when you try to cut. You will not be able to change the cut type without changing the design.
You can also identify an open path by moving the reticule over the endpoint. An open path will be indicated with a blue dot (see below).
If the shape was created on-tool, you can re-draw it with the pen tool, making sure that you return to your starting point before you tap 'Finish.'
If the problem occurs with an SVG that you created in design software, the problem may result from your file conversion process. Try returning to your design software and 'joining' any lines that you are trying to cut so that they are read as continuous, closed paths.
This process will differ depending upon your design software of choice. In Adobe Illustrator, for instance, you can highlight the problem area and use Object>Path>Join.