Not all users of the Embodiment Powers have to be a literal embodiment, being a representation of/representing something also counts. Athena and Ares represented the strategic and violent aspects of war, respectively
And here is what Power Manifestation, the power that covers all embodiment powers, says:
"The power to represent a certain power, quality, concept, emotion or elemental force.". The "Embodiments" powers covers all sorts of represetation of something, from simply, well, representing, to being a manifestation or personifying and embodying. Lots of embodiment powers also have users who representation or manifestation of something, and aren't the literal thing Tsubasa16 (talk) 23:03, November 24, 2018 (UTC)
Also, you say you don't find anything saying they are embodiments? then i guess you're not actually search and only saying what you know, here, taken directly from Wikipedia:
As i already explained, the "Embodiment" powers covers all sorts of represantation, the name "embodiment' is one conscientency as all the representation powers are called as "embodiment", the also called are also not always applied to all users, it's just alternate ways the powers can be called, just like some of the embodiment powers have "Incarnate/Representation/Manifestation" on the also called.
Uhhm, you are aware that this "Greekmythology wiki" is not a source, right? it is literally a fan-wiki about greek myth. Also, why are you even quoting the Riordian wiki??? Riordian series is a book series based around Greek Myths, IT IS NOT A SOURCE OF/ABOUT MYTHOLOGY, it is a book series BASED on greek mythology, you're linking it as if the Riordian series literally was Greek Mythology, and not a book series inspired by it, it is just a book series that has greek myth on it.
The only source you gave that is valid is Theoi.com. But even then, it is only one good source, just as wikipedia is as well, one thing being said in one but not on the other doesn't make it any less valid. Ares and Athena are personifications of aspects of war, just accept that and don't remove them again, in fact better avoid removing any Greco-Roman characters from any page, only adding, for that matter. Tsubasa16 (talk) 23:23, November 24, 2018 (UTC)
I wonder what part of "Representing also counts" you still didn't get. They may not be the literal embodiments of the concept of war, but they are representation of aspects of war, i already said that several times, they count, and period.
Giving me more sources that do not mention they being personifications will not change anything, i already said, "One thing being said in one source but not being mentioned on other doesn't make it any less valid", without also counting the fact Wikipedia is more reliable than any of the sources you gave.
And in fact, several of your sources say that Ares and Athena are representations??? one mentions Ares is the "unpleasant aspects of war/battle", other that "Ares is the Olympian god of war. However, unlike Athena, represents merely its destructive capacity and is typically the personification of sheer violence and brutality" on literally the very first page, are you even actually looking at your own sources? they countradict what you're saying, literally.