The Srimad Bhagavatam, I believe, illustrates King Parikshit (the successor of Yuddhistira) witnessing the coming of the Age of Kali in the form of a man beating a one-legged bull to death as a cow sheds tears. Parikshit realizes that the bull is Dharma, whose four legs are austerity, purity, charity, and truthfulness. The cow is mother earth, who is unable to withstand the sins of those incarning upon her. He then understands that the evil man is Kali, and that this all represents the coming of the Age of Kali (the Kali Yuga).
During the Kali Yuga, Kalki and Kali are supposed to meet in battle in order to decide the judgment of the world. However, I have seen nothing to suggest that the Kali yuga refers to the Age of Kalki and not the Age of Kali. I understand the linguistic points suggested, but have never seen any scriptural support for it. On the other hand, as mentioned above, I have found scriptural support for the Kali Yuga referring to Kali.