This is a misunderstanding.
The previous paragraphs talk about the requirements for the GPL and LGPL. Since the LGPLv3 is written as the GPL plus a set of extra permissions, you need to include both COPYING for the GPL and COPYING.LESSER for the LGPL additions.
In contrast, the AGPL is independent from the GPL and is contained in a single license text. So for an AGPL project you must not include the GPL license, and only the AGPL license text.
The “text version” in all cases means that the license should be included as a plaintext file, not in any markup format such as HTML, and not in any document format like ODT that would require special viewer software.
Per-file copyright + license headers are not required in any case, but probably still a good idea regardless of license. The value of these headers is that an independent file can be traced back to a project, and that essential license information is repeated in the file itself. However, the GPL licenses apply to a software as whole and not only to individual files, so as per my understanding only project-level license and copyright information is needed.