My understanding of the short ingeb version is that the woman, although originally abducted against her will and now escaped to her parents' castle, decides to rejoin her family down in the lake, thus departing from the land with its greenery. A simple story that does not essentially rely on supernatural effects; similar stories can be read in today's newspapers.
The variant with the failed divorce agreement seems to be inspired by good old Solomon. In the end, the lady caves in, although she may have contented herself with taking three kids, or sharing the fourth one on a monthly basis (difficult with the school, though).
I suspect the author was a man, whose message is that women will prefer to stay with their husbands and children even if abducted and deprived of air to breathe.