Truthfully, in all the restaurants we have been in, I have never seen any big deal made of "first visit" at a restaurant by Disney, other than a possible comment about hope you're having lots of fun, or asking what's your favorite thing so far? That said, if you want any sort of response, I'd say put it on ALL your reservations to increase the possibility that something special might happen...you never know.
For a birthday or anniversary, yeah, it's more likely something will happen. I have seen the Mickey confetti on the table (or the castle confetti at Cinderella's table), and special dessert presentation (like the chef spells out "Congratulations" or "Happy Birthday" in sauce on the dessert you ordered and paid for.) Sometimes a Mickey cupcake for the birthday person - but not always. But just as often the response ranges from nothing to the waiter telling you "Happy Birthday".