All transactions and balances are public knowledge; that's how the blockchain works. You can use a mixing service or send money to a dicesite or something then back to a new address if you don't want people to know how much you have, but you're never going to be able to truly hide your balances or transactions.
Thanks hilariousandco! Now the final piece of the puzzle, the last question unanswered:
I am left with 0.6 mBTC (in my Blockchain.info wallet) + 0.4 mBTC (in my online wallet) = 1.0 mBTC out of a initial sum of 2.0 mBTC, so half my money was taken by the miners in just one micro transaction?"If it's because of Blockchain.info's fees, I understand, though I thought the only fees when transmitting bitcoins were the 0.0001 BTC minimum recommended transaction fees (or sometimes 0.0002 if use an online wallet like Multibit). Is it possible Blockchain.info is adding something on top of these fees? (if so, they sure don't advertise it much to new customers like me).
TonyT