Some effort is required in order to protect your privacy with Bitcoin. All Bitcoin transactions are stored publicly and permanently on the network, which means anyone can see the balance and transactions of any Bitcoin address. However, the identity of the owner cannot be associated with their Bitcoin address until personal information is revealed by the owner during an exchange. This is why it is recommended for Bitcoin owners to use many different Bitcoin addresses; in fact, you should create a new one each time you receive money. This is especially important for public uses such as websites. You might also want to consider hiding your computer's IP address with a tool like Tor so that it cannot be logged.
The blockchain does not store any IP address. Read about the protocol or read the specs or read the code, or easier : ask anybody.
Bitcoin is absolutely not anonymous, and not intended to be anonymous. Which does not mean that the IP address of the sender and/or receiver is stored into the blockchain. The blockchain contains txid, amount, addresses and timestamps, but no IP address.
To examplify what bitcoin.org says in your quote : If you open an account on anybitcoinstoreorwalletorexchange.com, which assigns you a deposit address or which allows you to withdraw bitcoin to one of your bitcoin address, then this website will obviously be able to match your bitcoin address to the IP you used to visit the website, which might not be in your best interest depending on the website intention. Therefore, you could use Tor. That is all.
To the point of "a new one each time you receive money", it would be better phrased as "each time you receive money from a new untrusted peer". Let's say I got an account on example.com, and want to withdraw bitcoin from this website for the first time. It is considered best practice to create a new address to receive these bitcoin. Then whenever I want to withdraw from this website again, I could use the same address. This is the reason why bitcoin addresses can be labeled in the bitcoin software. Think of the tuple (sender_address, receiver_address) as a private key for a specific transaction funnel.
Bitcoin is not anonymous
Some effort is required in order to protect your privacy with Bitcoin. All Bitcoin transactions are stored publicly and permanently on the network, which means anyone can see the balance and transactions of any Bitcoin address. However, the identity of the owner cannot be associated with their Bitcoin address until personal information is revealed by the owner during an exchange. This is why it is recommended for Bitcoin owners to use many different Bitcoin addresses; in fact, you should create a new one each time you receive money. This is especially important for public uses such as websites. You might also want to consider hiding your computer's IP address with a tool like Tor so that it cannot be logged.