SQL Server's READ COMMITTED SNAPSHOT vs SNAPSHOT
READ COMMITTED SNAPSHOT uses a new snapshot after each statement. That means that less row versions are being kept alive. (The statement you quoted from the docs is slightly misleading because it suggest that this is always true - it is only true in case of long-running SNAPSHOT transactions.) Snapshot row versions are created on writes. The reads do not influence what gets put into tempdb. Writers cannot possibly foresee what reads will be carried out in the future. Readers only influence only what can be purged.
When a SNAPSHOT transaction T1 writes to a row that was modified by another transaction T2 in the time between T1 started and T1 attempted the write, the statement fails with an update conflict error. This is an optimistic concurrency model. With READ COMMITTED SNAPSHOT T1 would wait for T2 to release the X-lock on the row and continue normally.