Solved

Weather indicator is definitely out of synch between home page and setup page

Jack, which icon in setup page you mean? Because there are 2 of them

No, it's not, they are supposed to inform you that conditions have changed since your last practice lap.

Double checked with Jack (Basford) on this and yes, this is the correct answer.

I have another example of totally wrong weather indication. My next race is Italy and I wanted to do my setup.
The setup screen is showing dry weather:


Also the weather prediction screen shows the same dry weather:


But when I looked to the times of the other players, I saw that someone using wet tyres was much faster than someone with SS tyres and I knew both times were set in the past few minutes because when I looked 3 minutes before, no one had set a practice lap yet:


So I went to view an ongoing race in Italy and there I see it is actually raining with 3.6mm of rain:

and now the setup page shows rain, so there is a clear delay in between the weather that is used and the weather that is shown on the page.

Now the home page shows that it's dry and the setup page also. I do one lap and see on the time that it is still wet. Also now, after the run, the setup page suddenly shows that it rains. And although all indicators show that the lap was done in the wet, my driver says my setup is perfect (I used the dry weather setup from last season).
When I go back to the home page, it still shows dry.
When I go again to the setup page, it shows again sunny weather. When I do another lap, it changes the icon to rain again...
And in a live race I see it's still raining and 3.6mm.

There's clearly different delays between servers or race instances. For example in the old league the Rookie race had most of the time a delay of 10-15mins after the iGP page changed its weather symbol and the Elite race was another 10-15 minutes behind the Rookie race.

The real trouble about this is the practice laps are a complete mess. It might be that for each lap a different random server is selected with it's own delay for weather, instead, as would be expected, one with fixed values that match the ones during the race.

But even then things might still bug out. For example during todays race the weather got itself stuck completely. According to a league mate rain stopped on WU just after race start. As I joined in about 15-20 mins after the start both WU and the iGP page showed dry weather, but in both the Pro and Elite races the rain just continued unchanged for the whole race of almost 2 hours.

@Frank. I experienced the same at Monza tonight, it cost me big time. Maybe it would be a good idea to do what they did in the old game when weather was causing problems, turn it off until it is fixed.