Thank you all! I will most likely be using BC then. Sounds pretty safe and anonymous.
I will look into those other websites for another wallet.
If you want a case where your personal info could be found, here goes:
I own a Bitcoin exchange account that requires my address, name and everything to operate legally in my country.
I trade USD for Bitcoin on this exchange. My bank account is linked.
I cash out my Bitcoin to a random address.
The exchange can almost always assume that the address is mine, thus tagging this address with my address, name and any other information they know about me.