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Specifically, if we look at any transaction on the Ethereum blockchain, is the TimeStamp based on when the transaction is generated (like when you click generate transaction on MEW), or is it based on when the transaction was successfully sent from whatever node generated that transaction?

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As @Thomas has answered, timestamp of a transaction is the same as the block it is contained.As per my understandings, If I tried to explain why it's so;

is the TimeStamp based on when the transaction is generated (like when you click generate transaction on MEW)

If above is the case, transactions that get lesser priority in the pending transactions but was sent earlier would have a earlier timestamp but actually occurred in a later time will lead to a confusion of actually in which order the transactions took place. Because you can only be sure that a transaction happened, once it's mined in a block.

Let's say someone sends a transaction (say Tx-A) that at the point of sending it, with the account balance it was in a position to fail with an out of gas error. But due to low gas price while its pending the same account receives some Ether from another transaction (say Tx-B) which was sent later to the above mentioned one. Now the Tx-A comes to a position it can be completed without out of gas error while it's still in the pending transaction queue and later get completed. If the timestamps were the time it was initiated by the owner, that may look like the Ether from nowhere has been used in the Tx-A

And to make it sure, if you inspect any block's tx's at etherscan, you 'll see all of them have the same timestamp.

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The timestamp of every transaction in a block is the same as the block itself. All transactions in a block have the same timestamp. The timestamp is written by the miner that finds the block hash for that block.

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