First of all, the site you found is governmental, so a rather good find in the first place. It indicates:
Au restaurant, la carafe d'eau ordinaire en accompagnement du repas est inclue dans le prix du repas. En outre, les établissements de restauration et débits de boisson doivent indiquer la possibilité pour les consommateurs de demander de l'eau potable gratuite.
In the restaurant, the carafe of plain water to accompany the meal is included in the price of the meal. In addition, catering establishments and drinking establishments must indicate the possibility for consumers to request free drinking water.
The relevant law cited in the further article is Order No. 25-268 of June 8, 1967, which was repealed in 2016, but speaks about that such a charge needs to be included in the price. However, a different order concerning the display of prices from 1987/1990 is still active. This specifies in Article 5:
Les cartes et menus doivent comporter, pour chaque prestation, le prix ainsi que la mention "boisson comprise" ou "boisson non comprise" et, dans tous les cas, indiquer pour les boissons la nature et la contenance offerte.
Cards and menus must include, for each service, the price as well as the mention "drink included" or "drink not included" and, in all cases, indicate for the drinks the nature and the capacity offered.
Likewise, an environmental law in the shape of Article L541-15-10 demands the following:
A compter du 1er janvier 2022, [...] Les établissements de restauration et débits de boisson sont tenus d'indiquer de manière visible sur leur carte ou sur un espace d'affichage la possibilité pour les consommateurs de demander de l'eau potable gratuite. Ces établissements doivent donner accès à leurs clients à une eau potable fraîche ou tempérée, correspondant à un usage de boisson.
From 1 January 2022, [...] Restaurants and drinking establishments are required to indicate in a visible manner on their menu or on a display space the possibility for consumers to request free drinking water. These establishments must give their customers access to fresh or temperate drinking water, corresponding to the use of beverages.
Since the latter law actually requires them to give access to free drinking water (which is tap water) on request, they are allowed to not bring it out with the meal, as long as they have the required note that tells about it. However, many restaurants will deliver a flask of it on its own, sometimes even before you order.
As a side note: the serving of free water to meals is pretty much a custom in France, even if the waste reduction law of 2022 is very new still and the old law that had mandated it went away in 2016. When I last was in Paris in about 2005, I had to ask for water in one restaurant as they didn't deliver it on their own with the meal, upon which the server asked if I would prefer it chilled or warmed. In none of the about 10 different food places I visited in the week, water or table charges did show up on the bill, very much in accordance with the 1987/90 law. In one restaurant also the slices of baguette served while we choose our meals were off the bill, even though we asked for a refill of those.