[Ethereum] Lost ETH sent from Bitstamp

If I understand your question correctly you want to be able to see who has deposited Eth to your contract address. This is what event logs are for.

(1) Create a contract where there is an event every time there is a transaction. e.g something like:

contract someContract {   

    address public owner;

    // Set the owner of the contract to be the creator of the contract i.e. you
    function someContract() {
        owner = msg.sender;
    } 

    // This is an event 
    event DepositMade(address _from, uint value);
    event WithdrawalMade(address _to, uint value);

    //Catch all function 
    function() {
        // generate an event when someone sends you Eth
        if (msg.value > 0)
            DepositMade(msg.sender, msg.value);
    }

    // Only the owner of the site can withdraw Eth
    modifier admin { if (msg.sender == owner) _ }

    function withdraw(uint amount, address recipient) admin {
        if(recipient.send(amount))
            WithdrawalMade(msg.sender, msg.value);
        else throw;
    }
}

The important bits are defining an event type event DepositMade(address _from, uint value) and generating an event when something happens DepositMade(msg.sender, msg.value);these events are stored in the event log associated with the address of the deployed contract instance.

(2) You retrieve the events on this contract using rpc eth_newFilter or web3.eth.filter e.g something like:

var filter = web3.eth.filter({fromBlock:0, toBlock: 'latest', address: contractAddress, 'topics':['0x' + web3.sha3('DepositMade(hexstring,uint256)')]});
filter.watch(function(error, result) {
   if(!error) console.log(result);
})

Here's an intuition. The simplest transaction in Ethereum costs 21000 gas and looks like this:

{from: '0x from', to: '0x to', value: web3.toWei(1, "ether")}

When you're depositing to the exchange, the exchange can watch for transactions that are sent to this deposit address. For example, for block 99 they could use web3.eth.getBlock(99, true, ...), and for each transaction object check if to equals the deposit address, and the value (amount of Ether in wei) that was sent. Relatively straightforward for the exchange to see what you've deposited.

But if you are using a contract to make a deposit, the to is no longer the deposit address that the exchange gave you. The to is the address of your contract, and in the transaction there will be an additional data field. This data provides instructions to the contract, to execute code, which may then transfer the Ether to the deposit address. The exchange has to also execute the code, if it wants to be sure that it received funds from such execution. (An exchange could use web3.eth.getBalance to monitor balances of deposit addresses, but if it wants to know exactly where the funds came from, it has to execute the code of all transactions.)

That's an overview and to read more, the Ethereum Wiki has links to other material such as the Yellow Paper, and White Paper. Reading this StackExchange for highly voted questions and answers, with tags like transactions and gas is also helpful.

Technically, transactions are only initiated by humans: a contract invoking another contract is a message call (as defined in the Yellow Paper) not a transaction. Only the transaction is recorded on the blockchain because all further execution can be derived from the transaction.

For why withdrawing from an exchange to a contract wallet/account is not straightforward, it's because more gas is needed than the basic 21000 gas. Different contracts could consume different amounts of gas, and sufficient gas must be provided by the exchange otherwise the transaction does not take effect: the funds remain will remain with the exchange.