A nice fellow searcher has made me realized that I had not given any details about locations and languages used in the records... my mistake !
Actually my search has started here in France with a naturalization record which read Samuel as given name, born around 1885 in Biala Podlaska, Poland (actually stated "Russia" within the record). Then I found his death record, in Strasbourg, France, given name = Samuel, estimated birthdate 1883 (still same parents though...). Next finding, his marriage in Budapest, Hungary, given name = Sámuel Schmul, estimated birthdate 1884.
In Biala Podlaska records, I has been able to find his parents birth and marriage records as well as all? his siblings birth records, but no trace of a Schmul or Szmul or even cyrillic Шмуль in any birth record around the estimated date. The most probable would be the birth record of Szlama (translation on JRI Poland database of the cyrillic Шляма), born 1887.
Hoping this timeline would help.