In his Quantum Field Theory In a Nutshell, in page 12, (Second Ed), A Zee says that conventionally, the amplitude $\langle0|e^{-iHT}|0\rangle$ is denoted by $Z$. In the next paragraph, he considers the amplitude in eq. (6), and performs a Wick rotation to Euclidean time, $t\rightarrow -it$, and writes, $$ Z=\int Dq(t)e^{-\int_0^Tdt[\frac{1}{2}m\dot{q}^2+V(q)]} $$ which is the Euclidean Path Integral. I am confused because he denotes this quantity by $Z$. Is it the same $Z$ as $\langle0|e^{-iHT}|0\rangle$? If so, then I cannot see how. Can anybody help me?
I understand that $$ Z=\langle0|e^{-iHT}|0\rangle=\int dq_F\int dq_I\langle F|q_F\rangle\langle q_F|e^{-iHT}|q_I\rangle\langle q_I|I\rangle $$ with $|F\rangle=|I\rangle=|0\rangle$. How is this similar to the former integral?