Untrue. According to that same Kenyan Constituion, the change in citizenship was automatic.
Chapter 6 - Citizenship - Section 87
Persons who became citizens on 12th December, 1963
1. Every person who, having been born in Kenya, is on llth December. 1963 a citizen of the United Kingdom and Colonies or a British protected person shall become a citizen of Kenya on 12th December, 1963: Provided that a person shall not become a citizen of Kenya by virtue of this subsection if neither of his parents was born in Kenya.
2. Every person who, having been born outside Kenya is on llth December, 1963 a citizen of the United Kingdom and Colonies or a British protected person shall,if his father becomes, or would but for his death have become, a citizen of Kenya by virtue of subsection (1). become a citizen of Kenya on 12th December. 1963.
IF Zero was born in Hawaii, he would fall in the 'outside Kenya' provision in clause 2.
He and his father became Kenyan citizens on the same day.