What is the status of smart contracts in bitcoin? Are contracts described here already available for use? This source claims that "Ethereum style contracts" will be available in June. Is this correct? How are Ethereum style contracts different from contracts described in the first source?
Are contracts described here already available for use?


Yes.

Some of them are admittedly kind of arbitrary. For example, you can trustlessly sell someone a solution to a Sudoku. What does that mean? If you don't really have a solution, there is no chance that you will receive the money. If the other person doesn't give you the money, it's very unlikely that they will get the solution to the Sudoku. It's very interesting, but I can't think of a practical use for it.


  How are Ethereum style contracts different from contracts described in the first source?


Well, first, Ethereum script is a superset of Bitcoin script, so anything you can implement in Bitcoin script can also be implemented in Ethereum script.

Let's think of an example problem: suppose that you want to multiply six by seven, and send Bob some money if the result is bigger than 40.

There are two different approaches here:

The oracle approach would be that the oracle has access to a wallet with Bitcoins in it. The oracle multiplies six by seven, sees that the result is bigger than 40, and sends Bob some money. Nobody else on the network needs to know what was computed. They only check that the oracle's signature is correct.

The Ethereum approach would be to make a smart contract that has some money in it. Then, everyone on the Ethereum network multiplies six by seven, and everyone sees that it's bigger than 40. The smart contract sends Bob some money.

The weakness of the oracle approach is that the oracle could lie or cheat. The oracle could change the six to a five, and not send Bob any money. Or, the oracle could send the money to Eve instead.

The weakness of the Ethereum approach is that everyone on the network checks that six times seven is bigger than 40.