Let's say something was written to that volume after the snapshot was initiated. That data is not saved, but it will be saved on the next snapshot, right?
Follow-up question: What if the write corrupted the data because it happened just as the snapshot was being created, will all the following snapshots have the corrupted data on it?
Yes. The snapshot is at a point in time. Everything that is altered after that point will NOT be captured in the current snapshot but will be captured in subsequent snapshots.
Not sure how the write would corrupt the data. In theory, the snapshot operation is orthogonal on using the volume, so data corruption due to the snapshot should not happen.