How could we determine that a particular transaction in which block's Merkle tree?
As seen from the example image, we actually could not know that transaction-3 is located under that specific block and leaf-node since a light client only downloads the merkle root


As a light client, you would be given the branch/path of the tree which corresponds to the transaction you wish to know about, not just the merkle root. By giving you the transaction itself, and the branch of the tree, it lets you verify that the transaction was indeed part of that merkle root. Otherwise there is no way to tell, it is not possible to extra transaction IDs from a merkle root without being given the tree of transactions themselves. A merkle root is just a hash, a 'summary' and commitment to the transactions, it doesn't actually contain any information.