An Agile development process is one that observes the Agile Manifesto , that is: respect your values and apply your principles in pursuit of your goal .
And the goal is given by the first principle of the Manifesto:
Satisfy the customer through the early and continuous delivery of valuable software.
It may be difficult or even unfeasible to print all the values and principles of the Manifesto in the formalization of a process, so it is hoped that the result will be obtained from the assimilation of the Manifesto as a philosophy or culture , rather than respect for detailed formal processes. After all, the Manifesto itself "values more individuals and interactions than processes and tools."
This, which is the first value of the Manifesto, should not be used as an argument for not using any process, since the Manifesto itself was born from processes that were already used by its authors at the time.
The most famous examples of agile processes are Scrum and Extreme Programming , each of which is quite open and full of gaps that allow optimization for each environment each environment). Scrum focuses more on the project management level and XP delves deeper into software engineering by promoting agile practices as TDD and Continuous Integration .
There are, of course, many other Agile processes and every day someone else creates one.
You can also create your own. Strictly, it is enough to respect the values and apply the principles through practices and behaviors, focusing on "customer satisfaction through the early and continuous delivery of valuable software."
As for the detail of how the documentation should be, the Manifesto also talks about this:
We value more working software than comprehensive documentation.
Then your Agile process should look at this value as well.
In short , a requirement to implement an Agile process, whether market or proprietary, is to study and assimilate the Manifesto, accepting it in its essence.