As a beginner, practice with whatever paper you have lying around. Basically any paper you would throw into the recycling bin. Learning the fundamentals requires being comfortable with making mistakes along the way. You aren't going to appreciate the advantages of origami paper until you know what you are doing.

In simple terms, appendages are made from flaps. Multiple flaps can be created if you repeat the pattern of creases used to make a flap.

To illustrate this concept, think of accordion folding a sheet of paper to make a paper fan. Each rectangular ridge of the fan is a flap. It is not a very useful shape for a flap, because it is too short and too wide. Fortunately, there are various techniques in origami to make flaps in other shapes.

It is still origami because there is no such thing as purely clean folds. You opinion is based upon a semantic definition rather than a functional one. Paper might look 2 dimensional from a distance, but it is a 3 dimensional material.

Even the simplest origami models create layers of paper where nothing is truly flat. Paper itself has a tendency to flex and buckle at the creases. Many folders learn to adapt to the variance introduced by paper thickness.

What you have labeled as aggressive twists in complex models are not achieved by such a blunt method. They are flaps where numerous folds have converged. Like the accordion folds in a paper fan, there are many layers that add bulk to the appendage. You cannot haphazardly twist this to produce the final rounded shape. The creases concentrated in the area make it more prone to tearing.

There is a very vigorous and systematic process behind origami models of this level of complexity. Here is a video that explains the different levels of origami complexity.

https://youtu.be/MDwPXRy9IFc