The question is, with Rand and Lews Therin, do they have one soul or two souls in the body?
They have one soul with two personalities. The reincarnation of souls does not mean reincarnation of personalities. The personality develops with each reincarnation of the soul. This is the cosmology that I have cobbled together.
If you are viewing this on github.io, you can see that this site is open source. Please do not try to improve this page. It is auto-generated by a python script. If you have suggestions for improvements, please start a discussion on the github repo or the Discord.