Why is the Cult of Kosmos listed as predecessor to the Templar Order ? This is simply wrong and I think it should be removed. The predecessor of the Templar Order is the Order of Ancient and they've existed LONG before the Cult of Kosmos was even a thing. Both organisations have been "alive" at a same period of time and the Order's mission was already clear from the begginning so the Cult had no influence watshoever on the Order of Ancient/Templars.
Besides, they had a very thin alliance and I believe both sides would have been eager to get at each other's throat at the slightest mistake. I actually found a note from the Order saying that the Cult should not be made aware of their presence in Greece and another one saying "Amorges was right. I agree the time has come to purge the Cult from the Greek world."
It is possible for multi entities of Proto-Templar organizations to exist at or around the same time. The Cult believed in Kosmos, which in Greek, translates to "order" or "world", which is something the Templars pursuit. In Origins, the Order of the Ancients pursued the use of Isu artifacts and relics, which is also something that the Templars pursuit. You need to know there is approximately a span of 1500 years until the official formation of the Templar Order. Anything could happen, the remants of the Cult could have joined up with the Order down the road, or vice versa. There can also be various other Proto-Templar organizations that joined up before their official formation.
Maybe they joined up, but we don't know that. I think that Kassandra just totally wiped out the Cult and it's remains from the face of the earh. Saying The Cult of Kosmos is a predecessor to the Templar order/ Templar order future incarnation of the Cult is the same as saying that the Assassin Order is a predecessor to the Pirate Republic of Nassau because they both happen to believe and fight for individual liberties and chaos. It's wrong. It makes no sense.
It's like if I'm in Canada and I start a band. We call ourselves "The Creepy Zombies". I happen to form this band with some friends because we love to play music and later on we happen to take on students and share them our passion. Someone in India also happen to create a band of music and then later on take some students. They call themselves "The Rising Power".
We both love and produce music. We both share our passion. Then there is internal struggles and conflict arising in the band and "The Rising Power" just collaspse. The band is over.
My band in the other hand goes well and after some time we decide to change our name. We name ourselves "The Dark Wolves". Then we create a wikia page of our band. Now I will list "The Creepy Zombies" as a predecesor of "The Dark Wolves", but I certainly won't list "The Rising Power" as a predecesor of Dark Wolves/Dark Wolves as the successor of "Rising Power" as well just because we shared the same passion and Ideals, that makes no sense at all. "Rising Power" was its own thing and collapsed after a while and Creepy Zombies was another thing that evolved to become the Dark Wolves. Both bands had nothing to do with each other and had no influence over the other whatsoever.
To my knowledge, the Cult is not an antecedent to the Knights Templar. Their (admittedly contradictory) motives being similar to the OotA notwithstanding, there is simply nothing in-game which states that the Cult is also a predecessor.
The word "proto", according to the Dictionary of Cambridge, means "first, especially from which other similar things develop; original". The Order of Ancients already existed when the CoK was doing their things in the Greek World. In fact, the CoK was created around 6th–5th century BCE and the Order of Ancients was founded around 1334 BCE. Clearly, there's a difference. Also, the game states that Kassandra wiped out the Cult of Kosmos, this would be later confirmed by Pythagoras in the Cult Ending, the only survivor was Aspasia.
The only way I can see the CoK as a Proto-Templar Organization is considering that Aspasia ended up joining the OoA. She says that they [the Cult] will be replaced by a new kind of Order, one lead by a Philosopher King, probably in the name of the Father of Understanding or she was directly speaking of Alexander the Great (A Templar/OoA Ally) and that it was her task finding this new republic. I think she joined them but that needs confirmation.
Another alternative would be that the Order took something of the Cult and made it their own. Just like Darius, he wasn't an Assassin and the members of his group were killed on joined the OoA as well, but he's considered one because he created the Assassin's iconic weapon. Centuries later, in the Renaissance, he's posthumously honored with the title. The Children of Cain could be considered a Proto-Templar group, even when they were long before the OoA because they [OoA/Templar Order] took their insignia as they own when they became a public military organization in 1119 CE.
There are many groups that fight for freedom, but I think not all of them are necessary considered proto-assassins (unless they agreed to join the Hidden Ones/Assassin Brotherhood or they give something that helped the HO/AB in a way they could still exist). Same for the OoA/TO.