The Cowherd Star and the Weaving Maid Star have been famous stars in East Asian countries. Nowadays Altair is defined to be the Cowherd Star in public materials including science books, mass media, public outreach programs, and movies played in public planetariums and science museums. However, in astronomical books and star-charts in history, the Cowherd Big Star (牽牛大星) has been known to be Dabih or β Cap in the constellation Capricorn, and the asterism Cowherd consists of six stars including Dabih. Since the same title is given to the different objects simultaneously, considerable misunderstanding and confusion among the public have occurred. In this paper we consider this problem in two aspects. One aspect is which star has been defined to be the Cowherd Star in ancient literatures; the other is which stars were regarded as the Cowherd or the Cowherd Star in the historical records of occultation or conjunction in History of Koryo, Annals of the Choson Dynasty, and Daily Records of Royal Secretariat of the Choson Dynasty. As a result, we see that Altair has been regarded as the Cowherd Star in folklore and literature, but Dabih has been defined to be the Cowherd Big Star in astronomical works. An explanation proposed by previous researchers on the reason why such double meanings have been appeared is introduced. In the explanation, the fact that the Altair was defined as the Cowherd Star in ancient times had not been handed over properly, and the name of Cowherd was later put to the Dabih and its surrounding five stars to form one of 28 lunar lodges. Based upon these facts, we suggest the following ideas: (1) Altair should be introduced to be the Cowherd Star in public-friendly programs, and Dabih should be noticed to be the Cowherd Big Star in the field of history of astronomy. Dabih should be added as the astronomical Cowherd Big Star in academic books such as a dictionary of astronomical terminology. (2) The Korean pronunciation for Altair should be al-tear instead of al-tairu in accordance with the definition in the astronomical terminology dictionary compiled by the Korean Astronomical Society.