What does this mean? What do I do to get the previous owner of the coins I receive, with certainty?
If that person that sent payment to the Ping Service using a hosted (shared) EWallet, that "return" payment most certainly WILL NOT go back to the intended recipient. With a hosted (shared) wallet the Bitcoin address used for sending a withdrawal is not tied whatsoever to the account of the party requesting the withdrawal. That's why SatoshiDICE has a "Warning: please read first" on their site.
BitLotto maintains a list of clients that are valid (see "BitLotto compatible software"). It includes Bitcoin-Qt/bitcoind, Blockchaiin.info/wallet, Multibit, Electrum, Armory, Blockchain for iOS and Android, Bitcoin Wallet for Android, and others. There is one hosted Wallet that appears to not be shared, and that is Coinbase -- so that might be one EWallet option that would work with this PingService.