ST7789 TFT display only shows a correct image while Arduino reset button is held down - Why?
I found the answer to this problem. After reading multiple web pages / YouTube videos from multiple authors on this device and ~all of them warning you must never connect the Vcc to 5v or it will destroy the display, I found one or two that suggested you could use 5v because in this particular module there is a built-in regulator. So, faced with the thing being useless anyway, I decided to take the plunge and connect the Vcc of the TFT to the 5v pin of the Arduino and - hey presto - it works perfectly (see picture below: note the clear full image without the reset button being held down and also notice the voltage pin of the TFT is now connected to Arduino 5v compared to the pictures in my question).

The picture below shows the back of my TFT display module and, from what I have read, it is the J3Y transistor that is key to the voltage regulation in this module.

I hypothesise that pressing the reset button must have allowed equalisation of the voltages somehow but it is beyond my knowledge to explain it. I tried measuring the 3.3v pin on the Arduino with a voltmeter while holding down the reset button but I could see no change (the TFT module was not in situ for this test so maybe there is some interaction between Arduino and the TFT circuitry during reset).
Please note: I suspect this is a peculiarity of the particular TFT display module I have. Please DON'T run your 3.3v devices at 5v unless you have good reason to believe that they can take it. I only tried this after reading a couple of pages (including the detailed specs on the TFT module seller's listing) that suggested it would be OK with my particular TFT module - your's may be different. But if you have the same problem you see in my question then maybe that is something to consider.
Thanks to those who have responded.