Some are kaolinite, montmorillonite, and illite and they do indeed help with the plasticity. I am not sure if you can create clay from pure materials without specialized machines.
I am not an expert in this field but its one of the rare things that I remember from school, mostly because it was very relevant when helping my dad build a chicken coop.
Basically you need clay soil, which as far as i am aware is just a natural occurring soil layer containing those nutrients.
Depending on the local area that soil may already feel like clay and might be usuable as is. But its probably better to refine this into proper clay at home.
If you want to make your own clay I recommend simply asking around where to find clay soil. Almost every terrain anywhere exists out of multiple layers of different soil and chance is high that some people can tell you exactly what your local composition is.
The same nutrients that make clay are also very good to grow food, people have been studying these compositions for ages to decide where to start farming and building.
For me all that mattered was if dig a hole to put a pillar for the coop, i first had to dig trough normal dirt. Then had to struggle my way trough the “high quality clay soil which the local farm area is famous for”. To get to the much easier to penetrate sandlayer (still mostly dirt but the opposite of plasticity)