For clarity, you should say that you aren't counting Specialized Agency States (Kosovo, Cook Islands, Niue), because they are recognized by the UN, they just aren't members or observers. Or you could just add them, since it is "UN Recognis(z)ed Countries of the World..."
False. No such thing as "specialized agency states" exists, and besides that they aren't members of most or any UN specialized agency organizations. The UN as well as a large number of UN members do not recognize the mentioned regions as countries, thus they should remain omitted.
The United Nations recognizes each "country's" ability to participate in Specialized Agencies, plus more. The UN recognizes the Cook Islands and Niue as having full treaty-making capacity, and they are sometimes invited to join conferences, and Kosovo acts as a sovereign entity in certain contexts with the UN.