How to remove a repeated (not duplicate) node present in a same way?

I am working on importing a number of ways obtained as a result of vectorizing of a raster map. Because of the nature of the source data and as a result of chosen curve smoothing algorithm there are many cases when a single way intersects itself. In general, self-intersection has no automatic resolution in e.g. JOSM and requires human intervention, which in my case would be many hundreds of edits.

However, in my case it is always a very specific type of self-intersection that should rather be described as a "self-touching way". Let me illustrate it.

On the image above pay attention that there is a node an the intersection point, at direction of individual segments make it so that the right part of the "eight"-like figure touches the left part of it in that point. So it is not two duplicate notes (different IDs but same coordinates) but rather a single node (the same ID) that is present two times in a same way. Inspecting the OSM XML confirms this observation.

An algorithmic solution that is possible in this case is to remove one occurrence of that repeating node from the way. The result of such transformation for my example is below:

This transformation reaches the goal of preserving a "waistline" in the eight-shaped polygon, only making it slightly "thicker". This is an acceptable approximation given the resolution of the original raster data and noise already present in it.

The same effect can be achieved manually: a) duplicate the node, b) delete one of the copied nodes.

Such a solution is fine for my application and is guaranteed to create no new self-intersections because of the way the polygon is oriented (orientation of the top circle and the bottom circle always match).

I can come up with a solution that involves scripted editing of the underlying OSM-XML (see my own answer below). But I wonder if there are solutions to do it automatically without the need to export to OSM format, e.g. directly in the JOSM editor.

P.S. An alternative solution is to create two independent ways touching each other at the central node (upper circle and a lower circle). It is a bit harder to do when editing an XML in a sense that new ways have to be allocated, but is also acceptable for me if there are ready tools to do that.