That is correct, the initial save of blobs into storage would be a write operation in cool or hot, then switching to archive would be another operation:
When a blob is moved to a cooler tier (hot->cool, hot->archive, or cool->archive), the operation is billed as a write operation to the destination tier, where the write operation (per 10,000) and data write (per GB) charges of the destination tier apply.
When a blob is moved to a warmer tier (archive->cool, archive->hot, or cool->hot), the operation is billed as a read from the source tier, where the read operation (per 10,000) and data retrieval (per GB) charges of the source tier apply. The following table summarizes how tier changes are billed.
Source: MSDN