When the title of an article is printed differently in the front/back matter versus the first page, which should be used for citing the article?

Without sufficient reputation, I cannot post this as a comment; so here we go in an answer... I apologise!

'Beweis einer Baudetschen Vermutung' is orthographically incorrect, as 'Baudetschen' is an adjective and should therefore be written using a lowercase 'b' as 'baudetschen'. If this adjective, however, is derived from a name, it can be written as 'Baudet'schen' to emphasise it origin from the name 'Baudet'.

We can conclude that the title of the article itself is correct, whilst front and back matter are incorrect. The title of the book 'Nieuw Archief voor Wiskunde' does not look German to me (the cover lists Amsterdam and Groningen, two cities in the Netherlands), and indicates that is was not necessarily compiled by editors sufficiently proficient in German.

So this problem is actually not comparable to the potentially related question mentioned by OP and in the comments, as this book has a misspelling solely in its table of content, but not the article itself. Therefore I suggest to just quote the article by its correct title 'Beweis einer Baudet'schen Vermutung' and not to copy the editors' mistake 'Beweis einer Baudetschen Vermutung'.

Note the inconsistency.

The exact way of doing this will depend on the style or publication guide you are using. This idea comes from the APA Style Blog, where Timothy McAdoo suggests noting an unintentional typo using a footnote:

Linn, L. (1968). Social identification and the seeking of pyschiatric1 care. American Journal of Orthopsychiatry, 38, 83–88. https://doi.org/10.1111/j.1939-0025.1968.tb00558.x

1The published article includes this typo.

Similarly, if there are two forms of the title given, you could include a note that says something like this: "In the front and back matter of the journal, the apostrophe in Baudet'schen is absent from the title." Doing that, rather than messing with [sic] or assessing whether it's a typo, will give readers accurate information on the title as it is printed.

For a research manuscript with a digital object identifier (DOI), associated metadata may provide a title. Alternatively, and for research manuscripts without DOIs, a publisher's table of contents, index, etc. may provide a title. The actual title assigned by the author(s) may differ, and a publisher's usage may vary, hence, there's no "official" title. Ultimately, a citation is used to identify a source and the absence of an official title doesn't matter.