Check out the you tube channel ’interesting watches’, he has a few reviews of GMTs and his latest video shows the difference quite well. (I have no connection but I have just watched his video)
One type (eg Rolex) the hour hand can be adjusted independently of the minute and GMT hand allowing quick time zone changes this is something known as a traveller GMT the adjustment normally happens in hourly jumps.
The other type (eg eta powered GMTs ) the GMT hand can be set independently in hourly adjustments. When you set the time the GMT moves at the same rate as the hour hand so this means you have to do a bit of fiddling to set the watch up when abroad - setting the time to local and than adjusting the GMT hand back to home time.
In terms of half hours adjustments, it can’t be done on the either of these types of watch although I am sure there are some expensive versions out there that manage this.
There is one other type of GMT that can manage half hour changes and that’s the type that is found in the flightmaster, it is independent of all hands and you can adjust it as you want, it shows time in the 12 hour format.
With regard to use there is no correct way, you use it in the way that suits you.
When abroad I normally set the GMT to London time and the hour hand to local.
When in London I set the GMT hand to what ever time one I might be tracking.
I hope this rambling helps