Is the Archimedes screw some type of turbine?

Good question. We call it a turbine but it really doesn’t fit into one of the two main classes of water turbines, those being impulse turbines and reaction turbines. Here’s why;
1. Reaction turbine blades (runners) are fully immersed in water and are enclosed in a pressure casing. The blades are angled so that pressure differences across them cause them to rotate. The Archimedes screw is probably closest to this type of turbine but it is not enclosed in pressure casing. Like the Archimedes screw, reaction turbines are generally used on low head applications. Examples include: Francis, Propeller and Kaplan turbines.
2. Impulse turbine runners operate in air and are turned by jets of water hitting the runners. The Archimedes screw bares no resemblance to this type of turbine. Impulse Turbines are used with high head systems and use nozzles to produce the high velocity jets. Examples include: Pelton and Turgo turbines.

The Archimedes screw is closest in character to a Reaction Turbine because the flighting is actually in water but in some senses the screw is not actually a turbine since there is no circular spinning and swirling of water. Instead the space between the screw threads can be thought of as buckets of water and the weight of these buckets of water turns the screw.