Why does the imported JPEG from my Canon Powershot G9X Mark II on PC screen differ quite significantly from the camera display?

LCDs on different devices are different.

• The camera LCD is much smaller with higher dpi than the laptop screen. It's more difficult to pixel peep. Noise is often not visible on tiny screens.

• The brightness of the camera LCD is likely much greater than that of your laptop.

• Wider gamut? More likely with cameras that have OLED displays. The specs don't look special (3.0-type TFT color liquid crystal), and I'd expect Canon to make a big deal about it if it were.

I should've shot that in RAW and figured out the shadows later

That's a controversial topic. For a daylight scene with good lighting, RAW is likely unnecessary. You can consider RAW+JPG to have the RAW as a backup when needed.

Is adjusting photo settings for the next shot based on how the Live View screen displays the result of the previous shot the correct way of finding the right settings for a given shot?

You cannot trust the way the image appears on LCD. You have to evaluate the histogram and how the images appear on your intended output devices.

Can I see the same picture on the PC display and if yes, how?

1. Set your laptop monitor to full brightness.

2. See other questions about color-management

3. Adjust camera settings to produce images that will look better on your laptop. They're usually labeled color, contrast, highlight, shadow, dynamic range, etc. There may also be different color profiles you can try.

4. Adjust levels or curves to bring up the shadows.