It depends on the amount of rain and on the surface you are running/walking on. Now, there is unquestionably a threshold where the amount of rain makes the time exposure factor so significant that running wins without any doubt.
However for lighter rain there might be another factor that turns the result.
I remember reading a news paper article about running versus walking in the rain with text from NTB (and as far as I remember the experiment was conducted by NTB itself as well) maybe 10-20 years ago, where the conclusion was that walking made you least wet, based on weighing the clothes before and after the experiment. Unfortunately I am unable to find any references to it now.
Searching for it, I found many references to mathematical calculations like for instance here, which presents the assumptions for the calculations
There are, as you pointed out, two sources of wetness - the rain
that runs into you (on your head and shoulders), and the rain that
you run into (on your front).
and this is not correct, there is also a third source of wetness - water thrown from the ground with you shoes. And for this factor running is much worse than walking, which also was the conclusion from the NTB experiment.
Of course running in urban areas where the rain does not penetrate the ground is much worse than running on grass for instance.
So for heavy rain running is best (ignoring the risk of falling), for lighter rain it depends and walking might be the best option. The fourth source of wetness, sweat, I do not think is very significant compared to the other three.