>I don't know about the others but Evian and San Pellegrino each come from one source, respectively situated in Évian-les-Bains, France and San Pellegrino Terme, Italy.

Yes, that applies to San Pellegrino (and most probably Evian also) but San Pellegrino (the company, which is anyway Nestlè) has 6 bottling plants in Italy:

https://www.sanpellegrino-corporate.it/it/stabilimenti-59

While it is true that San Pellegrino brand water is exclusively coming from San Pellegrino (and as well Acqua Panna comes from Scarperia only and Levissima comes from Cepina Valdisotto only) the other three plants are anyway marked as "Nestlè Vera" (with of course a distincion on the spring, respectively "fonte in Bosco", "fonte Santa Rosalia", "fonte Naturae")[1].

I believe (possibly I am wrong) that in many (non-EU) countries the bottled water is more like a "pure brand" than a "specific spring".

[1]See page 63 of this .pdf:

https://www.sanpellegrino-corporate.it/files/valorecondiviso...