I finally answered my question about how to register
safely, i.e., relying official links only. In order to help some of you, here is what I found.
In order to register, go to
https://eos.io/ and click on the button "GET EOS", almost at the bottom, and right below the rotating logo. This button is not only for EOS token distribution, as one may figure out from the webpage design. After checking all the boxes, you will access to a page where you can buy, register and claim (see section "How to Participate"). Click on "register", or choose the "register" tab in the section "Participation Instructions". I let you read from there.
Interestingly, if you choose the MyEtherWallet option, you can generate a pair of public and private EOS keys. No need to use some third-party website as many advice, which I find very dangerous.
Using MyEtherWallet, you will issue a transaction with the "EOS – Contribution" contract. It should not take more than a few minutes if the transaction is sufficiently filled with gas. Take care, it is written and said everywhere that the "Gas limit" should be above 90000 and you will be fine. This may not be true. You need to have a gas price high enough, which is set at the top of MyEtherWallet pages. I used 20 Gwei, which is probably higher than necessary.