A pseudonym is any time you are using a name other than your own. A pen name is when an author publishes under a different name. A nom de plume is when a pretentious author publishes under a different name. Generally a pseudonym has the intention of concealing who the real person is. With a pen name or nom de plume it may be for purposes other than concealment.
When do you use 'nom de plume' vs. 'pen name' vs. 'pseudonym'?
@Mehrdad, isn't "Ham and Bacon" a nom-de-plume used by a single person? Therefore "Ham and Bacon suggests"
Was your wife's maternal grandmother from New England, perhaps? The earliest reference to "independent as a hog on ice" that I've found comes from Short Patent Sermons (1841), in a sermon entitled "On the Increase of Nominal Saints". The book is by Dow, Jr - allegedly a nom de plume of Elbridge G Paige This is introduced by a quote from Dryden: Truth is, our land with saints is so run o'er ...
Every time someone tried to reach him by landline (this being before cell phones were invented), his parents would find themselves calling out, "Zeno! Phone!" And that's how the author of Anabasis got his nom de plume.
I usually use out-of-quotes for phrases. For an extreme version of punctuation within quotes, see this 1915 statistics paper which mentions a pseudonym-using writer called "Student" with quotation marks as part of his nom de plume. Examples in the text include "Student's", "Student,", "Student." and even "Student†".
As various comments indicate, it could also mean a nickname, stagename, alias, nom-de-plume, etc. In fact, to different people in different times and places, second name can mean just about any name or part of a name that's not the first name.