What is the oath that necessitates expiation, and is it obligatory to inform the teacher of the truth for the forgiveness of the sin, if the oath was related to performing a school assignment and did not cause harm to anyone, with doubt regarding the wording of the oath?

The oath for which expiation (kaffarah) is required upon breaking it is an oath sworn by one of Allah's names or attributes concerning a future action, whether to do it or not to do it. As for an oath concerning a past event, there is no expiation for it, but rather a sin if the person was lying, and this is called a "Ghamus oath" (a deeply sinful oath).

Therefore, the oath that was sworn does not require expiation, and the matter should be considered from the perspective of whether or not there is sin. If you used equivocation (tawriya) intending a meaning other than the apparent meaning to avoid lying, then there is no sin upon you.

Regarding informing the teacher, the greater good (maslaha) should be considered; if informing her would bring about a benefit or avert harm to your studies, then it is preferable to inform her; otherwise, it is all the same.

Swearing and taking an oath mean the same thing.