I've always believed and heard this to be the Martin Luther translation of the German Bible. While I've not yet found a completely authoritative statement on this being the case, I currently believe it to be true. I think we may need get into his journals to probably get to the root of this. It may be possible that someone has written a research paper on this within the BYU libraries, but I have yet to put my finger on it.
One leading example is from Joseph Smith, Jr.'s journal (Journal, December 1842–June 1844; Book 1, 21 December 1842–10 March 1843) wherein the footnote references the more correct Martin Luther’s translation of Matthew 24:2 whereas Joseph refers simply to it as the "German bible." This tacitly gives us a link:
...not one stone shall not be thrown down.— German bible says which
shall not be broken...
 Martin Luther’s translation of Matthew 24:2 reads: “Es wird hier nicht ein Stein auf dem andern bleiben, der nicht zerbrochen
werde.” (Die Bibel, Matthai 24:2.)
Die Bibel , oder die ganze heilige Schrift des alten und neuen Testaments, nach der deutschen Uebersetzung D. Martin Luthers. Halle
(Saale): der Cansteinischen Bibel-Anstalt, 1826.
Granted this is not a direct statement, but I think the Martin Luther translation was considered to be the gold standard and most printed version for the German Bible at this time.
Martin Luther is again referenced in Revelation, 9 March 1833 [D&C 91] (Re: Apocrypha); See Historical Introduction:
Martin Luther, in his 1534 translation of the Bible, however, placed
the books of the Apocrypha at the end of the Old Testament and
Again by extrapolation it puts him in and around the "right place at the right time."
It also appears that Luther's translation was directly used in some other translations and/or influenced other translators, so at the end of the day Luther is the prime mover of the German bible, as per the Wikipedia article to which you referred.
Finally, in one of your quotations above, the author of the wiki article at this site attributes the German translation directly to Martin Luther (yet I see no actual authoritative citation for his source; perhaps basing it off of "common knowledge" such as my sentiment?):
Joseph Smith said of Martin Luther’s German translation, “I have an
old edition of the New Testament in the Latin, Hebrew, German and
Greek languages. I have been reading the German, and find it to be the
most [nearly] correct translation, and to correspond nearest to the
revelations which God has given to me for the last fourteen years.”
(Teachings of the Prophet Joseph Smith, pg. 349)
JBH adds an interesting observation in his answer that I thought I'd consolidate into this answer and add a little to assist the content:
Val Hinckley Sederholm's blog notes [Archive]:
I'm intrigued by the Prophet's polyglot New Testament consulted in the 7 April 1844 general conference address (King Follett Sermon) and in a 12 May 1844 sermon. "A Bible in various tongues" is how visitor Josiah Quincy described it. For some Saints the book became an iconic way of remembering Brother Joseph. A lithograph depicting the April 1844 general conference shows the Prophet with his Testament on the pulpit; the book even surfaces in Wilford Woodruff’s dreams: "I met with Br Joseph Smith in the Congregation of the Saints. He had his old Hebrew and Jerman Bible, and preached to the Saints." As the dream continues, Joseph, "thronged by people," lifts a curtain into "another room and there he was going to teach the people" (19 August 1844; 2: 449, in ed., S. Kinney; Josiah Quincy, Figures of the Past).
Note what Brother Woodruff calls the book: "his old Hebrew and Jerman Bible."
Thomas Bullock reports Joseph as saying: “I have an old book [Clayton: ‘N.T.’] in the Latin, Greek, Hebrew, and German” (7 April 1844). The book comprises “the old German [Luther’s original], Latin, Greek, and Hebrew translations” (Bullock Report, 8 April 1844). The Prophet further terms his Testament “the oldest book in the world” (Woodruff Journal and Clayton Report for 7 April), which matches Samuel W. Richards’s characterization of it as “an ancient German Bible” (12 May 1844). Only one polyglot Testament fits the description and that is Elias Hutter’s Novum Testamentum harmonicum (Nuremberg, 1602). (emphasis by JBH)
(Elias Hutter can also be found as Elia Huttero (aka Latin: Hvttero) (Archive.org search) and each available scanned book may be viewed here: Matthew, Mark, Luke, John and Romans
– These were made available November 25, 2014, while the blog post was from March 2, 2010 so these works were not available online at the time of blogging.)
All the arrows, so far, seem to converge on Martin Luther...