Should the error term of PID be normalized?
If e(t) and u(t) are nondimensionalized then Kp is unitless, Ki has units of time^-1 and Kd has units of time. Otherwise they have units from the process.

In the second formulation you've made a mistake. This is the one to use.

$$u(t) = K_c\left( e(t) + \frac{1}{T_i}\sum_{t}e(t) dt + T_d\Delta e(t)dt^{-1}\right) + K_k $$

Then Kc has units from the process and Ti and Td have units of time. This is good because Ti and Td are on the same scale and only Kc has process units. e(t) and u(t) do not need to be normalized because just Kc relates their units.