To avoid the T-intersections you can shorten all lines a bit by using the Extender and a small negative value for the extension. Then use the Intersector (output Node) to generate the intersection points.
I agree with @erik_jan, use the Extender with a small negative value before the LineOnLineOverlayer:
Just remember that the extension length is in ground units and that you need to find a suitable value base on your dataset. E.g. if the coordinate system of your dataset is in meters, then -0.01 equals one millimeter, etc.
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