Is “How Cai Yong was condemned for mourning Dong Zhuo” true?

Houhanshu (Biography of Cai Yong, part 2) says he only sighed and changed colour while seated at Wang Yun's, and died in prison rather than being strangled: 「及卓被诛，邕在司徒王允坐，殊不意言之而叹，有动于色。」「邕陈辞谢，乞黥首刖足，继成汉史。」「邕遂死狱中。允悔，欲止而不及。」 Chapter 009 of Romance of the Three Kingdoms rewrites this as Cai Yong throwing himself on the corpse exposed in the marketplace: 「董卓暴尸于市，忽有一人伏其尸而大哭。」「原来那人不是别人，乃侍中蔡邕也。」 His plea to be allowed to finish the Han history does match the history: 「愿公见原：倘得黥首刖足，使续成汉史，以赎其辜，邕之幸也。」