It looks like a bug. If there is already a vertex at the intersection point on the line to be split, the lines will indeed be split. However if there isn't, the line remains intact.
It is particularly problematic for lines that touch each others, as you see the vertex for the end of the line but the intersected line doesn't always have a corresponding vertex.
Let's note that is happens even if you snap your points, as snapping ensure proper alignment only, it doesn't create extra vertices.
'Split Lines with Lines' requires two layers, the input layer and the split layer (which I think you have though it's hard to confirm with your images). The description of the tool states:
This algorithm splits the lines in a line layer using the lines in another line layer to define the breaking points. Intersection between geometries in both layers are considered as split points.
Your network isn't intersecting, but touching. Try the tool again but with the splitting line layer overlapping the 'Input Layer'