I'm neither an etiquette expert nor a psychiatrist, so maybe I shouldn't be offering anything here, but I found myself being a little sympathetic toward you, based on your question.
I can see how she would be offended, and I don't think it was the "greatest thing in the world" to do; but (maybe I'm wrong, I know) I'm picturing that you had the card with so few words and wanted to "turn it into a son card" (like the kind that come with "to Son and His Wife" on them. I'm imagining how then, because you're a parent, it was just kind of automatic that you threw in "Joe" I (because none of those "to our son" cards are really personal enough for the relationship we have with our kids, but we overlook that if we're not the ones writing on the front of the card).
I'm picturing that because you were "designing" the front of the card as a way of adding something that would "turn it into" a more special "son and his wife" card; it was natural that you'd feel the need to throw in "Joe" but not as natural to include his wife's name, because you have a different relationship with her (and it just didn't come naturally to include her name the way it did with you son's name).
To me, your mistake wasn't so much not including her name, but in not thinking when you were "overcome with the urge" to add Joe's name. I'm also guessing that your wish to make the card more like an "official" son card (and therefore more special) was the primary thing on your mind, and your natural instinct to include your son's name (because it seemed to impersonal when it came to him) "just happened". In other words, I think you were so concerned with making the card a "son and his wife" card you weren't thinking. I'm also guessing you liked the "live, love, laugh" verse so much you wanted them to have it; but maybe, again, wanted to make it a "son and his wife" card.
I can see how Joe's wife (whatever her name is :) ) might be offended, but then again it wouldn't kill some people to lighten up and realize sometimes people just don't think (especially in the "hurry" of filling out holiday cards etc.). (Oh well, I'm guessing you won't make that particular mistake again. Good luck ironing out any hard feelings.)