The use of zero should be a clue that "Ubaid 0" was identified later in the timeline of Ubaid archaeology, and was an attempt to fit an earlier phase into an existing structure. Eridu was first excavated in 1855, 1918, and 1919, then Tell al'Ubaid in 1919; but Tell El'Ouelli wasn't excavated until 1976-1989. The name "Ubaid period" was adopted at a conference in 1930.
There is a more relevant later article than the one you cite. The Wikipedia article on Tell El' Ouelli cites the following source:
Huot, J.-L. (1985), "Tell el'Oueili. Principaux Résultats de la Quatrième Campagne (1983)", Paléorient (in French), 11 (1): 119–123, doi:10.3406/paleo.1985.4367, retrieved 2 August 2011 (http://www.persee.fr/doc/paleo_0153-9345_1985_num_11_1_4367)
where we can see, in the English translation of the article's abstract:
Main results of last season (1983) at Tell El'Ouelli near Larsa (south Iraq). At the beginning of a sequence covering the well-known Obeid period 1-4, a new phase, provisionally called Obeid O or Ouelli Phase, was discovered.
So, occupation of this settlement in south Mesopotamia preceded Eridu, but continued through the rest of the Ubaid period, up until some time before 3000 BC. Quoting the Wikipedia article:
The excavations have revealed occupation layers dating from Ubaid 0 (6500-5400 BCE) to Ubaid 4.
The actual details are in the French text of the article, but the authors were clearly convinced that it fit into the structure of the Ubaid period.
Sous les structures, apparaissent en suite des murs associé materiél du type Eridu, ou Obeid 1.
Under these structures, material associated with walls of the type Eridu, or Obeid 1, subsequently appears.
Eridu gets pre-eminence because it continued through the Uruk Period, and then as a religious site through many later periods of Mesopotamian history. But Ouelli shows that it wasn't really the beginning. We have to relegate the notion that Ubaid was solely the spread of ideas from Eridu to a theory that had to be re-evaluated based on later findings.