Do all of these mean the same thing? If not, what’s the difference?
Cutlery is, effectively, sharp knives and related implements (e.g., the large two-pronged fork that accompanies a carving knife). It can mean a set of “steak knives” and thus overlap with flatware.
Flatware comprises the pick-up-and-use portion of table services, i.e., knives, forks, and spoons. Silverware is, obviously, flatware made of silver (with silver plate usually included).
Eating utensils is the broader term including place settings and serving utensils. (E.g., one does not give each person an olive fork, a gravy ladle, etc., but places one in or adjacent to the foodstuff with which it is to be used, to be made use of to serve food to each person at the table.) It can also include the oddments that accompany certain foods, such as the little pronged forks with insulated handles that are placed into the ends of corn-on-the-cob to facilitate handling the ears of corn in some circles.
Essentially, the terms define distinct categories of things-used-in-preparing-and-consuming-food with some overlap between them.