It sounds like we agree about the important parts here, but to clarify those where my statement wasn't clear enough:
"Neo4j did not create their own license": They did. They stuck some legal terms together and said "this is the license". They didn't do it well, and they might have done it so not well that their license doesn't let them do what they intended, but they did make a license document.
"they did not ‘have the power’ to add terms while masquerading as AGPL. Note - they kept the AGPL name, FSF copyright, and preamble - that clearly states the AGPL has to be verbatim."
This is where it is unclear. For one thing, they didn't masquerade it as the AGPL. Before any text, they state what they think the license is:
The software ("Software") is developed and owned by Neo4j Sweden AB
(referred to in this notice as “Neo4j”) and is subject to the terms
of the GNU AFFERO GENERAL PUBLIC LICENSE Version 3, with the Commons Clause as follows:
Are they allowed to do that? It is unclear, and the FSF don't want them to, and when the FSF told them not to, they complied. However, that is not the same as masquerading since the modification was stated up front. Neither is it clear whether they can be challenged by the FSF on the verbatim part, since they do include the AGPL verbatim, then tack their clause at the end. As I said in another comment, if I write an introduction to a book but do not modify the content of the original book, you wouldn't say I had edited it, so it isn't clear whether the FSF can or has chosen to exert control over anything else that appears.
In addition, these details only are relevant to a case, which has not been filed, where the FSF sues Neo4J over misuse of their trademark and/or violation of the license to their license. It does not answer what that potentially misused license means. Even if the FSF is correct about all of this, all it means is that they could make Neo4J change it, which they have done voluntarily. It wouldn't necessarily mean that you can apply other terms to the software. Whether you can apply those terms is a different issue, and the chance that maybe you can only exists because Neo4J did not stick their license together well and ended up with something contradictory.