Most sources state that Joseph was born on 1st. December 1892
in (or near) Poszony, Hungary.
However, in April 2014 a Rootschatter found a baptism record from 1890 (!) for Josephus Remenyi, who was baptised in Trenčianska Teplá, Trenčín, which is about 140 Kilometres N.E. from Bratislava. In this record the parents are given Franciscus Remenyi and Johanna Mayerk.
And as possible 'circumstantial evidence' ? : I also have a photo of his parents, on their wedding day, and of a very young Joseph, taken by a professional photographer from "Trenchin & Tepplitz" [Trenchin = Trenčín] (see below, and in parents' biographies).
I also have the baptism certificate of his sister Maria, who was born in 1893 in Žilina, which is a further 50 Km. away. In this certificate, the parents names are given as Frantizek Remeny and Johanna Majer.
Shortly afterwards, another Rootschatter discovered Joseph's application for Naturalisation in which he states quite clearly:
I was born in Trencsn Teplic Hungary ... My last foreign residence was Porsony Hungary;
(Poszony is the hungarian name for the city known in german as Pressburg, and which is now
in Slovakia and is called Bratislava).
Joseph married Margit Papaloczy in 1918.
She "americanized" her name to Margaret.
Joseph Remenyi died on 25th. September 1956, Cleveland, Ohio, USA.
On Monday Sept. 24. Professor Remenyi dismissed his class complaining that he did not feel well. He was taken to the Lutheran Hospital in Cleveland. He suffered a heart attack and passed away on Tuesday morning, September 25. 
"If it had not been for that story I might have become a railway official like my father and spent the rest of my days saluting the chief of the railroads."Instead, he published his first book at nineteen and became a reporter.
Certificate of Arrival--For Naturalization Purposes
US Dept of Labor, Immigration Service
Ellis Island, NY, Mar 17, 1920
This is to certify that the following-named alien arrived at the port indicated, on the date and in manner described below, viz:
Name of alien: Remenyi, Josef
Port of entry: New York, NY
Date of arrival: May 11, 1914
Name of vessel: Carpathia Cunard Line
He was married to Margit [americanised: Margaret] (whom I never liked very much, she never had any time for us kids). They had no children. I do remember him visiting several times during our summer holidays in the Salzkammergut. He was a wonderful storyteller.
"He finished up as Prof. of Philosophy at Western Reserve University, Cleveland, Ohio. There was a statue of him, (perhaps just a head ?) somewhere in the University. which was knocked down and ruined during riots in the sixties...."
"where his lectures on European writers often earned spontaneous applause from his classes", rising to the rank of Professor in 1949.
The site of the Hungarian Garden was dedicated on October 21, 1934, upon the occasion of the 123rd anniversary of the birth of Franz Liszt, with the unveiling of the bas-relief of the Hungarian composer. [...] Speakers at the dedication included: ... Hungarian Consul Joseph Remenyi, ... www.clevelandmemory.org/ebooks/tpap/pg61.htmlAccording to another source, he worked for the Hungarian Legation in 1914, but I haven't seen any mention anywhere that he was the Hungarian Consul.
On July 23, 1950, at the conclusion of the annual One World Day celebration, marking the 25th anniversary of the founding of the Cultural Gardens, a bronze statue of Imre Madach, philosophical dramatist and author of "The Tragedy of Man" was dedicated in the Hungarian Garden. Dr. Joseph Remenyi delivered the principal address on the works of Madach. www.clevelandmemory.org/ebooks/tpap/pg61.html
Many thanks to all the RootsChatters, who added various missing "bits" to this story.