In addition to the stages mentioned, Piaget redefined knowledge by suggesting that some people get stuck in certain stages of development. For instance, a mentally disabled person may not advance to the abstract thinking stage. This leads to the idea that people can have intellectual ages as well as chronological ones.
Piaget redefined knowledge by determining that (1) knowledge is developed in four invariant, hierarchical and universal stages and (2) children are not cognitively able to perform some tasks of logic and deduction, which academic opinion assumed they could perform, until they reached age 11 or older. The first stage is infancy to two years of age. The second stage is two years of age to seven years of age. The third stage is between seven years of age and eleven years. The fourth and last stage begins at eleven years of age or older and continues until after adolescence.
The first stage (0 - 2 years old) is the sensorimotor development stage in which the infant is believed to understand the world only through physical actions within or upon the physical world. Actions in stage one are founded upon inborn instincts that are utilized in expanded tasks. The second stage (2 -7 years old) is the perceptual or intuitive stage which follows language acquisition and in which events are believed to become independent of physical action and result in mental images.
The third stage (7-11 years old) develops concrete operational intelligence during which class and relationship comparisons emerge and recognition of multiple salient characteristics supplants object recognition based on single salient characteristics (e.g., if a hot-dog looks like more when cut up, then it is more). The fourth stage (11 years and older) develops formal operational intelligence in which logic, reasoning, deduction, hypothetical thinking and thoughts of time beyond the present emerge. Piaget's theory asserts that stage four components of formal operational intelligence are not possible for children prior to the invariant, hierarchical, universal fourth stage, which is a concept that redefined the previous assumption that children could engage in logical, hypothetical, deductive reasoning at early ages.
You can basically ask what was Piaget's contribution to the field of education. Every specific element you mention in the question falls under the umbrella of Pedagogy (in my humble opinion the BEST field one could ever study). The answer to that much simpler question is that he was one of the first proponents of developmentally-minded tasking for students by levels of ability. Back in the day there was no way anyone would even consider the idea of differentiation based on skill. Back in his time, it was all rote learning, memory, and dry facts. His contribution was so huge that is still accepted today as one of the most effective ways to teach and one of the most productive ways to implement instruction at a general and specific level.
Piaget developed the idea of stages of development. This was primarily contrary to popular research which said that education was not a form of hierarchical types of learning but anyone could do anything at any time.