Confusion about names and name variants can lead to confused identities and brick walls.<sup>1</sup> I'm glad to see questions like this posed in context. 

You wrote, "How do you decide what is somebody's "real" surname?"

One of the best way to learn someone's real surname is to **seek out records containing primary information about the name**. For example, **forms and/or documents/letters the person filled out, authored and signed**. Elizabeth Shown Mills describes this as "standard usage" about those who are/were "literate" and where "signature samples exist." <sup>2</sup> 

Depending on the record circumstance, the marriage license about your relative and her second husband might include signatures/this kind of primary information. There might be deeds and/or letters, too.  You mentioned that he immigrated from Croatia; possibly there is a naturalization record about him that contains his signature. 

By comparison, we don't really know the circumstance about how most names would have been enumerated in 1930 US census (probably an oral communication; unidentified informant). As well, the name given for the decedent on a death certificates is provided by a third party, too. 

Ha! If *she* acted as the informant on *his* 1932 death certificate, then her signature might appear therein. 

Notes and References:

 1. One of the best ways to learn about usage practices is to read articles in scholarly genealogical journals, such as the *Register* or *Quarterly*.   A recent article focused on how learning more about a difficult name and its variants helped to solve a genealogy brick wall. See the *Register* article by Henry Z. Jones, Jr., titled, "The Ancestry of Mary Whitten (Whiting), wife of Ichabod Crippen" [begins at vol 166 (Oct 2012), p. 259] From the article, "By checking various indices of surnames of  colonial American families and seeing how “Whiting,” ‘Whiton,” “Whiten,” “Whittin,” “Whetten,” “Witon,” “Whitten,” and even “Whitney” often were used in the seventeenth and eighteenth centuries in reference to the very same name, a new avenue of exploration as to the ancestry of Mary “Whiting” Crippen opened up. This article is the result of that journey."
 2. Elizabeth Shown Mills, "[Re: \[TGF\] Surname spelling variants][1]," Transitional-Genealogists-Forum-L message dated 24 Jun 2011.  


  [1]: http://archiver.rootsweb.ancestry.com/th/read/transitional-genealogists-forum/2011-06/1308944480