Step 1: Understand the time sequence.
The clause
\[
\text{``By the time the manager arrived''}
\]
describes an action that happened in the past.
The completion of the presentation occurred before the manager arrived.
Step 2: Choose the correct tense.
When one past action is completed before another past action, the earlier action is expressed in the Past Perfect Tense.
Structure:
\[
\boxed{\text{Subject + had + Past Participle}}
\]
Step 3: Analyse the options.
• (A) completed -- Incorrect. Simple Past does not clearly show which action happened first.
• (B) has completed -- Incorrect. Present Perfect cannot be used with a finished past time.
• (C) had completed -- Correct. It indicates that the presentation was completed before the manager arrived.
• (D) was completing -- Incorrect. Past Continuous indicates an ongoing action, not a completed one.
Step 4: Final conclusion.
\[
\boxed{\text{By the time the manager arrived, the team had completed the project presentation.}}
\]
Hence, the correct option is \(\boxed{(C)}\).