QGIS: Using split lines with lines makes unwanted cut on self-overlapping lines

I have some long lines that get through some points. I need to cut those lines each time they pass on one of these points.

I managed to do it using the graphical modeler but here comes my issue.

In my model I use split lines with lines tool. There, I cut my long lines with smaller ones I have created on each point. But I discovered that when a line I want to cut overlaps itself, it gets cut on this point and not only when it gets through a point.

So here is my point, is there a way to make "split lines with lines" not splitting my lines that overlaps themselves. Or have you got any idea of how I can manage to deal with this.