Did Berzelius unify the names of chemical elements using the Latin language?
What I have found from the literature is that Berzelius's proposal helped unifying the symbols of chemical elements, but the names remain diverse in language sources (not all are Latin) and some remain disputed (not all unified). Thus, I gather that Berzelius's proposal helps, but it is far from the statement that he unified all the names of chemical elements using Latin.


This is essentially true. Berzelius was the first to suggest using symbols abbreviated from the names of the elements and this system, with some modifications, is still used today. Most of Berzelius' symbols were abbreviated from Latin names but a large number of the elements discovered since then have had symbols drawn from their common names and some of Berzelius' original symbols have been dropped in favour of more modern versions. Dalton had earlier used a system of drawn symbols to represent the elements but this was not widely taken up as the symbols were difficult to draw and memorise. (source)

Looking though this wikipedia article on the etymology of chemical elements, the most common language of origin appears to be Greek (argon, chlorine etc.), with Latin coming in second (mercury, rhenium etc.), including quite a large number of names which come originally from another language (often Greek) via Latin.

Additionally there are a large number of elements which are named after places (Swedish - ytterbium, yttrium, etc. ; English - americium, californium, etc. to name just a few) or people (mendelevium, einsteinium, fermium etc.) and have been given the Latinate endings -ium or occasionally -um (see this question for more discussion). Also note that according to IUPAC recommendations:


  For linguistic consistency, the names of all new elements should end in “-ium”


The common, and IUPAC recognised, names of gold and gilver both have old Germanic origins and are recognised as official names because of their widespread usage. I have never seen the Latin names for gold, silver or lead (aurum, argentum and plumbum) used in a serious chemical context, although of course the elements' symbols (Au, Ag and Pb) are derived from them. 

As such I would not recommend using the Latin names as the main recognized name in a Wikipedia page, although a side note about them might be justified.

EDIT: Thanks to @KlausWarzecha for pointing this out. There is one exception where the latin names (aurum, etc.) come into play: in anionic complexes, the latin version of the name is officially used. $\ce{[Cu(CN)4]^{3-}}$ is tetracyanocuprate(I), $\ce{[Ag(Cl)2]^-}$ is dichloroargentate(I), $\ce{[Au(Cl)4]^- }$ is tetrachloroaurate(III). However, this does not change the viewpoint that the Latin names should not be used when commonly discussing the element.