An Agile development process is one that observes the Agile Manifesto, that is: respects its values and applies its principles in search of his goal.
And the goal is given by the first principle of the Manifesto:
Satisfy Customer through Early and Continuous Delivery of Valuable Software.
It may be difficult or even impossible to print all the values and principles of the Manifesto in the formalization of a process, so it is expected that the result will be obtained from the assimilation of the Manifesto as a philosophy or culture, more 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 not to use any process, because the Manifesto itself was born from processes that were already employed by its authors at the time.
The most famous examples of Agile processes are the Scrum and the Extreme Programming, each of them very open and full of gaps that allow them to be optimized for each environment (or worsened for each environment). Scrum focuses more on the level of project management 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 each day someone creates one more.
You can also create yours. Strictly speaking, just 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 software running than comprehensive documentation.
So 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.
I believe it is "history" - http://wp.clicrbs.com.br/sualingua/2009/05/06/a-triste-historia-estoria/
– gmsantos