If you initiate a transaction, wait a few hours, and the transaction doesn't appear in block explorer, then the coins weren't sent. Placing a transaction in the public hash chain is essentially the definition of "sent" in this context.
Exactly, for Mt. Fun and I MyBitcoin is deducting funds from our accounts but not initiating a transaction to the recipient.
Exactly correct. The funds sent appear in the block explorer and arrived in Mybitcoin. Then I sent them from Mybitcoin to another wallet and they never got there and the transaction is not in the block explorer but Mybitcoin records the payment as sent.