Do I register my greeting card under an "artistic" or "literary" copyright in the U.S.?

Probably neither, it probably doesn't make sense to register a copyright to a greeting card, even though I'm sure it does have both literary and artistic elements. You could simply affix copyright notice to the back and claim common law rights to it, but I wouldn't bother filing an application for registration, especially if you've created lots of greeting cards, which could get very expensive.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

You've posed a false choice.

When registering the copyright in a work of authorship, the work must be characterized by the author as either a (1) Literary work, (2) Visual arts work, (3) Performing arts work, (4) Motion picture or other audiovisual work, (5) Sound recording, or a (6) Single serial issue. See < http://www.copyright.gov/forms >.

Of these, only the literary or visual arts categories can apply to your greeting card.

It's a bit existential, but nonetheless true, that all physically-existing works of authorship (including literary works) can be visualized. The "visual arts work" category, therefore, is not the repository of all works that can be viewed but is, instead, restricted to those works that can ONLY (or at least predominantly) be experienced through sight.

Because you note that your greeting card is text only (fancy text, perhaps, but still only text) I see no dilemma here. Your greeting card is a literary work.

Whether it makes sense to register the copyright in your card is a different question. For my money, if you
make the card publicly available (through sales or free distribution) and think that someone else may copy and start selling the card, then I think it makes sense to pay the $35 and take an hour or so to fill in the online registration form. If you don't, or if it's unlikely anyone will swipe your card, then there's no reason to register its copyright. It's always a good idea, however, to place the copyright notice on all of your works of authorship: Copy. 2009 [author]. All rights reserved.

Have fun.

If the text is a poem or has literary meaning, then literary.
The remaining elements are just one of many possible expressions and cannot be claim to exclude others.