I think the other party refers to the one who cast it.
The original kanji is "津久葉家の誓約は、仕組みでいえば相手に自分の望む魔法を使わせる術式だ" I think the closer translation would be "In terms of how it works, the Tsukuba family’s pledge is a spell that compels the other party to use the magic the caster desires."
I also find that very suspicious because Tatsuya is technically strong but as everyone knows that he's incapable of doing much complex magic because his magic calculation area is filled with regrowth and decomposition. Saegusa should've known this since yokohama incident and infact she's the one...
I think the translation they used for the anime is pledge and its mental magic.
That's probably because the magic is hidden deep within Miyuki mind and Tatsuya need to dig deep without any other interruptions so sniff it out.
Most likely notified by edward clark as part of his scheme to get rid off tatsuya. The guy do have god complex and think that anything that his country and by proxy him, is the only power that has the right to control the world.
Or they just want to adapt everything as soon as possible since only relying on one mangaka will take longer than multiple, either way its just really odd decision.