There is a paper I want to cite where, on the title page of the article, the authors names appear without any middle initials (i.e., one of the authors is Mark Smith). However, this one author frequently publishes using his middle initials (i.e., Mark E. Smith). What is the appropriate way to cite this-with the name as it appears in the article, or the name as the author typically uses it? For what it's worth, the source article appear in a mathematical journal, and I want to cite it in a paper that will eventually appear in a mathematical journal.
You should use the exact information which is provided on the published paper, not the real or current information of the author; even if his name or affiliation1 is change at the present time.
Moreover, the correct way of citation of each paper is provided in the webpage of its publisher and you can check how they have mentioned the author's name in the citation example (you may also download the bibtex or other outputs to be used in citation manager softwares). At least, by checking the publisher website, you will be sure how they prefer their published paper to be cited (even if other possibilities/doubts for correct citation exists).
1I know that affiliation is not mentioned in citation, but if somebody wants the affiliation of the author at the time of publishing the paper, the best source is the information written on the paper and is provided in its publisher's website.
You need to cite articles the way the name is expressed in each article. The purpose of the reference if for others to be able to locate the source you have used and so making the reference accurate is essential. It may seem like nit-picking since most of the reference will be correct except for one initial but it is better to simply follow the the generic rule to follow the article in all details than to modify it. One instance where this may matter is when article references are listed in, for example, Web of Science. There an article may end up as two entries if there is a difference in the way it is referenced. I have seen and personally have articles that have multiple entries because of this and because people wrote the wrong year, volume etc. Still, the importance lies in being able to trace your sources and being correct makes that easier.