Citing full names of authors Different journals have different standards regarding author names - some mention only the first name (i.e. "William Webb"), some mention also an initial of the middle name ("William A. Webb"), some mention only initials ("W. A. Webb"), etc.

So, when I copy BibTeX records of papers to my bibliography manager, the names are copied in different styles, and also appear in different styles in my bibliographic listing.

My question is: is this a problem? Should I go over all my bibliography, each time I write a paper, and make sure all names have the same format?
No, you shouldn't change your bibtex database every time. Bibtex converts automatically from its native format "Surname, Firstname I. and Other, Author" to whatever the journal style is (for instance: F Surname, A Other), as long as you have full names in the database. Name formatting is something that people often get wrong at first when they have little experience with bibtex.

So you should aim for having full names in the database, at least for those papers that you need to cite with an unabbreviated first name. Everything would be easier if all journals used initials, but it's a tough world.

I suggest initials for theses, preprints and submissions, and only switch to full names after the article is accepted in a journal that demands them (and after being asked by a copy-editor that demands them). This is the solution that minimizes the workload on the author.