The three levels
Processed means the transaction was seen by the node you asked. Confirmed means it landed on a block that the cluster has voted on. Finalised means it sits under enough stake that reversing it would require an extraordinary event. Each is a stronger promise than the one before.
Which to wait for
For a read of your own balance, confirmed is usually enough. For a transfer you must not lose, wait for finalised. The mistake is treating processed as success — it only tells you one node heard the transaction, not that the cluster agreed it landed.