While preparing a bibliography, I have noticed that some authors' journal titles have not used diacritics correctly. For example, some authors have written "Miguel Angel Asturias" in their titles rather than Miguel Ángel Asturias.
Should I be referencing these titles exactly as they have written it, or should I correct them?
In my opinion you shouldn't change the title.
The purpose of your reference is letting your readers find the referenced work, so changing the title (even if it is to correct a mistake) may make this harder or even create inconsistencies in scientific indexing services.
To add to the answers already posted: authors themselves sometimes avoid diacritics, presumably because they (used to) complicate indexing and search as well as increase the likelihood of citation errors. A colleague of mine decided to forego diacritics in his papers and on his Google Scholar profile for this reason. So if you suspect that might be the case you'd be better off not changing anything.
The situation is different when a reference to an old paper has been mangled through a citation chain. This happens fairly often with author names and can make it a nightmare to track down the original paper. And even if it doesn't, it is sometimes obvious that the paper is indexed incorrectly - for example, the first and last names of authors are inverted. These cases call for a correction although that doesn't seem to apply here.
This is really opinion based. I am happy my surname contains one letter beyond basic ASCII table...
The reasons I purposely omit the accent are:
Imagine what can go wrong if your surname is Štěpánek (Stepanek, %C5%A0t%C4%9Bp%C3%A1nek). Please, try to read aloud the name before following the link to Wikipedia, where the pronunciation is shown.
From the cited author's perspective, I'd be happy to be cited using plain ASCII charset. My name won't be crippled by encoding issues. I'd be also happy when my name will be dispayed in same way through whole document, the correct way preferably.
From the referencing perspective, the thing that really matters is that the reader can find the cited resource properly. DOI, ISSN, ISBN, Journal tags etc. are the true keys to follow when looking for the references. Author Name(s) and Article Title fields are "optional extras" for the reader. I've seen many articles where the reference contained only the first author name and the Journal Tag (for example: S.H. Sheng et al. / Acta Materialia 61 (2013) 4226–4236).
My advice is:
This is, perhaps, a judgement call. I would correct them. There may be "reasons" why the original wasn't accurate, even if it is just not knowing how to produce the letters of an expanded alphabet on your keyboard. I suspect, but don't know, that the diacritics might actually change the meaning in a few cases. But in this case it is proper to "honor" the person named even if it is a bit out of sync with the one doing the citing.
But others might disagree, believing that, when quoting, it is necessary to be precise.
A solution, of course, is to put the corrected name (or other word) in parentheses, following a quote. Or, some would write ... Angel [sic]... to indicate that the original is being preserved even though incorrect.
I'll also note that search engines seem to be sensitive to this issue and find the item correctly. For example searching Google or Duck Duck Go for "Gabriel Garcia Marquez" behaves correctly even when the name is in actual quotes. It also finds items that don't correctly accent the name. Likewise a search that has the proper accents finds items that don't.
Should I be referencing these titles exactly as they have written it, or should I correct them?
Look for those authors' websites; other published work; and especially works they published all by themselves. If the lack-of-diacritics is an outlier, I'd say add them; otherwise leave them out.
