Wet bars fall under a different GFCI requirement
The issue of whether a room is a "kitchen" or not is only marginally relevant to GFCI requirements, as NEC 210.8(A) point 7 requires GFCI protection within 6' of any sink in a dwelling unit, including those found in wet bars:
(7) Sinks — where receptacles are installed within 1.8 m (6 ft) from the top inside edge of the bowl of the sink
However, your question has merit
However, your question does matter as it determines whether your space requires its own small appliance branch circuits. For that, we turn to the NEC Article 100 definition of "Kitchen":
Kitchen. An area with a sink and permanent provisions for
food preparation and cooking.
A wet bar has a sink, of course, and also has "permanent provisions for food preparation" in that it has countertop space. However, the lack of provisions for a permanently installed cooking (food heating) appliance is what makes it "not a kitchen" under the NEC.
So, adding a built-in cooking appliance such as a cooktop or hardwired oven would do the trick. What else would make it a kitchen? A "hutch" for a microwave or toaster oven or other such provision obviously intended for a cooking appliance to be left connected/installed 24/7, would also push it over that line. Simply plugging a hotplate/portable induction cooker or other such portable cooking appliance in and leaving it on the counter wouldn't be "permanent" enough in my eyes, though, since someone could readily unplug it and put it away if they needed to use the counter space for something else.
Of course, nothing in the NEC stops you from dropping a pair of dedicated 20A branch circuits to the countertop receptacles on your wet bar, even if there isn't a microwave hutch or other "kitchenette" feature planned there.