In early 2009?
The only way to mine bitcoin at that time was to run BitcoinQT and the result of that would be for the Bitcoin to be mined into an address associated with that instance of BitcoinQT. That would NOT be a transfer.
If we throw away all the pieces of your story that don't match reality (assuming that you're memory has just gotten a bit distorted over time), the best that can be pieced together from what you're saying is:
Either they convinced you to install BitcoinQT on your own computer, and convinced you to run it to mine some bitcoin for yourself. Or they ran BitcoinQT and mined the bitcoin to their OWN wallets. Then they could potentially transfer that bitcoin to you, but you'd have needed to supply them with a bitcoin address for them to have done that. If they were running a website that they convinced you to log into and that website showed a bitcoin balance, then you almost certainly never had any bitcoins. Those bitcoins (if they even existed) remained under their control.
You claim to have sent them $10?
Did you use PayPal? Did you mail them a $10 bill via U.S.P.S? Did you give them your credit card number? Did you use Venmo? Did you go to a local bank branch and deposit $10 into their bank account? Did you use Western Union?
Too many of your statements fail to match up with well-established history. This leaves a few possibilities:
1. You are attempting a scam of some sort, but didn't spend enough time verifying the story you created.
2. Your memory of dates and events has become clouded a bit over time, and you are unwilling to accept that possibility
3. You have come from an alternate reality. You travelled across dimensions and are now in a parallel world.