Difference Between Integral Time and Ki(Integral Gain)
There is no hard rule about the terms Integral time and Integral gain.  It is up to individual manufacturers to decide what version to use, mostly based on custom or end application.  In general, things like heater control use the term integral time (or 'Reset'), and faster operating devices like motor speed controllers will use the term integral gain.

For a parallel PID equation, using time as the integral term, the term is represented by 1/time, so the larger you make time, the slower the integrator will react.

When gain is the integral term, it is used to multiply the integral term with a fixed time period, so the larger you make gain, the faster the integrator will react. Then the term essentially is the sum of Gain * 1/time, but the time term is not adjustable.