TL;DR: Rikka probably taught Nibutani the incantation before she headed off to battle Dekomori. The incantation probably does not come from Shichimiya, since Shichimiya uses a different incantation in her battles against Rikka in season 2. Moreover, Nibutani most likely never did battle as Mori Summer before, so it makes sense for Rikka to have taught her the incantation to start her battle with Dekomori.
Firstly, I dispute this piece of evidence:
If I remember correctly, Sophia Ring SP Saturn VII also uses the same incantation. This is understandable since Sophia Ring SP Saturn VII was friends with Dark Flame Master and went to the same school if I remember correctly. Tyrant Eye came into existence due to Dark Flame Master.
The first time we see Shichimiya battle is in season 2 episode 3 Tsuigeki no… Maou Mahou Shoujo ("Magical Devil Girl… in Pursuit"), when she challenges Rikka at the latter's school. The incantation Shichimiya uses at that time is different from Rikka's. The subtitles I've seen is translated as
"Sing, cherubim! Descend, seraphim! Physical linkage!"
Perhaps someone knowledgeable in Japanese can identify the precise incantation; however, it is doubtless a different one from Rikka's, especially as the scene cuts between Rikka and Shichimiya as they each deliver the corresponding lines of their incantations.
The same thing happens the second time Shichimiya and Rikka battle, this time in episode 9 Namiuchigiwa no… Kyuukyoku Ougi (Rizouto Rasuto Rizouto) ("Ultimate Mystery… on the Shore (Resort Last Resort)"), when Shichimiya confronts Rikka at the beach about the latter's resolve.
However, in the movie Take on Me, when Shichimiya and Rikka are about to battle at Aomori, Shichimiya responds to "Reality, be rent!" with "Synapse, break!" and both chant "Vanishment, this world!" together, before their battle is stalled by Kumin. But this instance can be chalked down to the director's creative decision; I would say that it's clear from the instances in season 2 that Shichimiya usually uses a chant different from Rikka's when going into battle.
Moreover, while it is true that Yuuta received his name "Dark Flame Master" and his attack "Be engulfed in the flames of darkness and disappear!" from Shichimiya, as he reveals in episode 3, he does not mention anything about the incantation. In fact, I think the only instance when Yuuta prepares an incantation of any sort is when he shows Rikka the Unseen Horizon right at the end of season 1. In that instance, he used Rikka's incantation, but I would hesitate to say that that shows that he was familiar with Rikka's incantation before meeting her and getting to know her at the new school. So, while it is true that the Tyrant's Eye came into existence due to the Dark Flame Master, who in turn came into existence due to Sophia Ring SP Saturn VII, I don't think this provides any link between Rikka's incantation and Shichimiya's incantation.
How then does one explain Nibutani's use of the same incantation as used by Rikka and Dekomori in her fight with the latter? My reading of that episode is as follows.
Recall that Rikka was present when Yuuta was about to offer his advice to Nibutani about how she can get Dekomori to hand over the Mabinogion. The scene cuts away at this point, but it is reasonable to assume that Rikka was present during Yuuta's suggestion to try and defeat Dekomori in battle. I surmise that Rikka might have stepped in at some point and taught Nibutani the incantation that she and Dekomori use to enter battle.
After all, Nibutani has possibly never done battle before during her phase as Mori Summer. In season 2 episode 3, when Rikka is practicing her attacks against Kumin—who is playing the role of a provisional magical girl—ostensibly to defeat Shichimiya the next time they battle, Nibutani interrupts them and points out that Kumin is more of a sentou shoujo ("Battle Girl") than a mahou shoujo ("Magical Girl"), and she goes on to add:
In the first place, magic is supposed to make people happy. The spirits who manipulate magic were always thoroughly pure of heart. They never resorted to brute force…
Her explanation is dropped midway, but her non-violent approach as Mori Summer is clear, not only from the above explanation, but also from her general demeanour and actions as Mori Summer: having priestess-like accessories, singing choir songs, talking to the birds, communing with Nature, etc. All of these point far away from a battle-oriented chuuni persona.
So, my guess is that when Yuuta guided Nibutani to face Dekomori in battle, Rikka offered Nibutani her incantation to kick things off.
