There's the third case: they care. In which you wouldn't be able to tell whether the card is "genuine" or AI written; the two things aren't even meaningfully different in this scenario.
You can tell if the thank-you card is hand-written. Most people don't have a pen plotter connected to their AI text generator to write thank-you notes.
Emailed or texted "thank you" notes don't count. At all.
Yes, but they could also generate the text and transcribe it onto paper by hand.
For many people, myself included, 90%+ work on things like thank-you notes, greetings, invitations, or some types of e-mails, is in coming up with the right words and phrases. LLMs are a great help here, particularly with breaking through the "blank page syndrome".
It's not that different from looking up examples of holiday greetings on-line, or in a book. And the way I feel about it, if an LLM manages to pick just the right words, it's fine to use them - after all, I'm still providing the input, directing the process, and making a choice which parts of output to use and how. Such text is still "mine", even if 90% of it came from GPT-4.
I guess if someone went through the effort to prompt an LLM for a thank-you card note, and then transcribed that by hand to a card and mailed it, that would count. It's somehow more about knowing that they are making some actual effort to send a personalized thank you than it is about who wrote it.
But honestly I don't think "blank page syndrome" is very common for a thank-you card. We're talking about a few sentences expressing appreciation. You don't really have to over-think it. People who don't send thank-yous are mostly just being lazy.
My financial advisor sends out Christmas cards and Birthday cards. They are pre-printed stock cards. I don't even open them. I should tell him not to waste the money. If he even wrote just one sentence that expressed some personal interest, then they would mean something.
These kinds of messages are on the one hand just pro-forma courtesies, but on the other hand they require that that some personal effort is invested, or else they are meaningless.