Question regarding collector leakage current with open emitter
The reason for this contradiction is that different definitions for the common base current gain are used in equations (1) and (2). In equation (1), \$\alpha\$  is the DC current gain defined as the ratio of the DC collector to emitter current \$\frac{I_C}{I_E}\$ as measured at the terminals. In equation (2), \$\alpha\$ is defined as the the ratio of the change of collector current divided by the change in emitter current at constant collector-base voltage \$V_{CB}\$. It corresponds to the small signal common base current gain \$\partial I_C/\partial I_B\$ at constant \$V_{CB}\$. For practical purposes, the DC current gain (1) is often used to approximate the low frequency AC current gain defined by \$\alpha=\partial I_C/\partial I_B\$ because in good transistors their values are usually very close to each other.