当初 and 当時: an original plan and the situation then
Compare an original schedule or three-person plan with photographs and student status from a past time, identifying the time reference of each word.
An original plan as the reference
当初の予定より作業が早く終わった。 reports work finishing earlier than originally planned. 当初, read とうしょ, supplies the original plan as the reference for the comparison with the actual finish.
計画は当初、三人だけで始めるつもりだった。 says that the project was initially supposed to begin with only three people. 当初 concerns that initial intention; it does not supply the number who actually began or a later change in participants.
Photographs and student status then
当時の写真を見ると町の変化が分かる。 says that photographs from the time show the village's changes. 当時, read とうじ, points to the period shown in the photographs rather than an original plan to take them.
私は当時まだ学生で、その店によく通った。 places being a student and frequently visiting the store in that past period. Both details describe the speaker's circumstances then.
The initial plan versus past circumstances
当初 refers to an initial plan or thought, whereas 当時 refers to conditions at the time. The original three-person intention and the speaker's student status are different kinds of time-based information.
The earlier finish is an actual result. The three-person start is an intention, while the photographs and student status provide past background. Keep each statement within its supplied scope.
Are the initial three-person start and being a student then both original plans?
Show the answer and why
No. 当初 refers to the initial intention or plan. 当時 describes circumstances at that time, including being a student and frequently visiting the store.