The European Food Safety Authority (EFSA) which evaluates those claims has published a report on their decision.
Scientific Opinion on the substantiation of health claims related to water and maintenance of normal physical and cognitive functions (ID 1102, 1209, 1294, 1331), maintenance of normal thermoregulation (ID 1208) and “basic requirement of all living things” (ID 1207) pursuant to Article 13(1) of Regulation (EC) No 1924/2006
They conclude for the first claim:
The claimed effects are “hydration, e.g. body function, physical and
cognitive performance”, “adds to fluid intake and supports
hydration”, and “hydration”.
A cause and effect relationship has been established between the
dietary intake of water and maintenance of normal physical and
The following wording reflects the scientific
evidence: “Water contributes to the maintenance of normal physical
and cognitive functions”.
This means that you can advertise the mentioned claim on water bottles you're selling. In the appendix they list for claim 1331 the wording "water keeps you hydrated", which is pretty much what was mentioned in the article.
For the third claim, that water is a basic requirement for all living things, they concluded
The claimed effect is general and non-specific, and does not refer to
any specific health claim as required by Regulation (EC) No 1924/2006
I'm not sure if that means you cannot advertise that or that it isn't a medical claim and it doesn't fall under the regulations.
There probably is more history to the whole thing, I could imagine that there were some disputes about the exact terminology. The document is published in April 2011, so it was available when the article in your question was published.