The current full official names of the organization are simply Ezras Torah in Hebrew and Torah Relief Society in English as can be seen at http://www.ezrastorah.org/about_us.php and in their mailings.  

(Mossad) Ezras Torah was founded in August 1915 and was known by that name in numerous Yiddish and Hebrew news articles, advertisements and publications from the early 20th century.  Ezras Torah's own publications (see the links to the hebrewbooks.org documents on the wiki page) also mention just Ezras Torah without any addendum.  One can also see "Ezras Torah Fund" without any additional title listed in page 45 of the following PDF document American Jewish Yearbook, 1921-1922 (Directories).
The organization was incorporated in early 1920 under the name Torah Relief Society.  The incorporation document can be found in New York City's Division of Old Records in Manhattan.

I can think of three possibilities re: "for Relief of European and Palestinian Rabbis"
1) The 1936 volume by Julius Caesar Schwarz was giving a description and not adding to the title
2) That book was in error
3) Ezras Torah had a different official name in the mid-20th century (this would need to be supported by official documents or from material published by Ezras Torah)

If either of the first two possibilities were correct then the later English language publications listed above might have just copied the lengthened organization name from earlier English language sources without consulting more authoritative sources.