Should I cite author names as they appear in the journal or as I know them to be complete?

I disagree with the statement of "a publication must be referenced exactly the way it is published."

From Knuth's Mathematical Writing

In his bibliography Knuth has tried to keep his citations true to the original sources. The bibliography contains mathematical formulas, full name spellings (even alternative spellings when common), and completely spelled-out source journal names. (This last may be unusual enough that some members of a field may be surprised to see the full journal name written out, but it’s a big help to novices who want to find it in the library.)

This means that you can change reference, give full name of authors, give full name of journals. It may be unusual but it can be done and it is DONE. It is done by one of the most respected authors in the academy.

To answer your question:

So my question is: am I incorrectly citing the article when I include more complete information about the author's name than is given by the journal article itself?

When you include more complete information, you are not doing anything wrong. Actually according to Knuth, you are doing very good job.

Therefore if your target journal does not have any rule about same reference rule, you should be perfectly alright.

I can see your problem but I am afraid you will have to live with the mess. The reason is that a publication must be referenced exactly the way it is published. The exception is that it is permissible to abbreviate first names to initials to adhere to the standard of the publication in which you intend to publish. It is, however, not allowed to remove middle names (initials) if they are part of the original publication, nor to add initials if they were not part of the original. I know many authors who have published under one and two initials randomly, but there is not much one can do about it.

Now, if you do it in your reference program, there is nothing wrong with that, but you need to make sure your changes do not migrate into published work that you write.

Just an example, Uniform Requirements for Manuscripts Submitted to Biomedical Journals: Manuscript Preparation and Submission: Preparing a Manuscript for Submission to a Biomedical Journal states:

Some but not all journals check the accuracy of all reference citations; thus, citation errors sometimes appear in the published version of articles. To minimize such errors, references should be verified using either an electronic bibliographic source, such as PubMed or print copies from original sources. [The text then goes on to discuss retracted articles.]

You should not add an initial that was not present in the original publication or expand initials into full names if the author used initials. One reason is the one Peter Jansson gives, namely that the bibliography should reflect what was actually published. However, there's another reason I find even more compelling: respect for the author's choice of name.

Editing another person's name is presumptuous. Like many people, I have deliberately chosen the version of my name I use in papers, and I care about this aspect of my professional identity. What my passport says or what I'm called in person are irrelevant, and I would not be happy to have citations edited to use what someone else thinks is a better (or more formal, or more complete) version of my name than the one I chose to publish under. If my paper says Alice P. Liddell, then nobody has any business deciding I should be called Alice Pleasance Liddell instead.

In particular, your desire for completeness or consistency should not outweigh the author's autonomy to choose their own professional name, and you should not overrule a deliberate choice. Of course, some people aren't picky about what they're called, and they may publish using several random variants of their name. It can be tempting to standardize the name for the sake of consistency, but you should avoid doing so unless you know the author wouldn't mind. If you don't know the author personally, it's hard to distinguish between an author who doesn't care and one whose preferences have changed over time, and it's not reasonable to rely on your own guess as to how to handle this. (For example, if someone adds an initial due to marriage, they might be displeased to see it retroactively applied as if it had always been their name.)

There can also be political aspects of naming. For example, some women publish under initials to avoid drawing attention to their gender. I'm not convinced this makes a difference, but who am I to unilaterally undo it by replacing the initials with identifiably female names?

To summarize, names can be a sensitive subject. Every paper comes with the names chosen by the authors at the time of publication. If you're going to modify these names, you'd better have a good reason.