What could be the possible noun of the time between the minimum and maximum?

It sounds to me that you want to set the minimum and maximum times for a speech, and then have the yellow light turn on at the midpoint between the minimum and the maximum.

In order to use your app effectively, the users need a term for this time (in order to refer to it in conversation with others), and a definition (to explain what it means). The term will be used frequently. The definition hopefully less so.

You don’t need to use a mathematical term. You could just as easily use a functional term, such as warning time, and then explain the calculation in your definition. This would allow you to have a default calculation, which could be correctly described as the average, the midpoint, the half-way point, or something similar. It would also allow the organizer (I forget the official Toastmasters term) to have longer or shorter “yellow times” to suit the group’s taste.

I can understand your reluctance to use average time, since the term average is usually applied to a set of measurements that have been completed, whereas your application (as I understand it here) is more about indicating the stage reached in a process that is still continuing. Average is technically correct in describing the calculation, but semantically wrong in relating the calculation to its real-world use.

My suggestion for the term is caution time, since the yellow light on a traffic signal is typically referred to as a caution.