Just from my own perspective, the term "training" implies something I have not yet witnessed. The weight calibrations feel like exactly that, calibrations.
Calibration is a feedback loop of fitting your model to the data. In statistics, that's a bad thing. For ai, that is how they are trained. The end result is the calibration values.
If you calibrate your model on weather patterns or if you calibrate it on how words appear in texts, those will be different values inside your hard coded model, that now can do this hyper dimensional math things to predit what comes next. Like an answer to a question. Or if it rains tomorrow.
But anyways, it appears that nothing ever actually touches the network itself.
There is no memory. There's nothing to remember.
Yes and no.
The model itself is not changed by using it. It really acts like a black box transformator where you put in something and get out a response specific to that something.
And it does have a memory for your session, but actually it does not have a memory. Just as you said, the previous text is just fed in again.
I recently saw a good explanation for this memory thing.