I really want to do this as well someday. I love being in water but didn't like the idea of having a traditional pool with all the maintenance and especially all the chemicals. YUCK! When I saw the natural swimming ponds/pools I loved the idea. You get to have a swimming pool and have it be helpful to nature and not harmful. I saw one where they used rock gabions along the side to form the sallow area for the plants that help to filter the water. I can't remember how they made them exactly. I don't think they used metal wire. Because it seems like the metal cage would rust in the water over time I think they used wood post with wood slats cut from trees on their property. to hold them in. I think they already had enough large rocks on their property to do this. But you could always buy them. I would NOT use pressure treated wood even though some suggest that. You are just putting toxic chemicals in the place you want to swim and have plants maybe even fish if you want them. I know they did put some kind of pumping system in to keep the water circulating. But if your is spring fed and is clean water maybe you wouldn't have to. Not sure about that. Have you checked out
https://www.totalhabitat.com they have plans you can look at.
I hope you can get it done. Let us know how it goes.