Yes, PWS comes on above TLA 53 degrees, same microswitch as your TO Warning, but needs warm up. There is also alternate scanning involved, between use of the radar disc for weather and PWS, and alternate scanning for the Captain and F/O, leading to a possible delay of PWS detection of 12 seconds.
Given a 36 sec average time to rotation, the first third of your take off, when the PWS caution is active, you have thus possibly no indication.
I was kind a weary you do not have those reference books handy.
Nowadays most people I train, get their knowledge from SOP's and answering multiple choice questions. Times have indeed changed.
That does not mean I changed accordingly. I like to keep it simple, and have my WX radar switched on at Line up.
In our company we do not pay for the unavoidable extra fuel burn this habit creates.