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.
in the moment that you link your bank account, they can start search for your identity, so in theory even exchange that do not ask directly for your ID are not anonymous
better to not exchange bitcoin altogether if you want to keep your privacy or use monero with this