How to set the barometer as a weather indicator tool?

Pressures in Garmin terminology:
Ambient Pressure:
The true ambient pressure at your location, measured by the pressure sensor in the watch.
The value changes with the weather.

Barometric Pressure:
The equivalent pressure at Mean Sea level.
This value also changes with the weather, synchronous with the Ambient Pressure.
In barometer mode, the watch calculates Barometric Pressure from three variables: The true ambient pressure at your location, your known altitude and an assumed density of the air column between your altitude and Mean Sea Level.

You mention that you have a manual setting for sea level pressure in the watch. I do not have this setting in my Fenix 3. I assume that the setting is intended to be used in altimeter mode in the special case where you do not know your initial altitude, but you do know what the equivalent pressure at Mean Sea Level is right now at your location. But there is also a slight chance that the setting is somehow used for adjusting a calculation offset if your local weather station uses another air density for calculating the equivalent pressure at Mean Sea Level, compared to what the watch uses.

In either case, it is important that you do not just input some textbook average value for air pressure at mean sea level. The actual value changes all the time depending on the weather and your location. You need to use the actual value. If in doubt, I would not set anything.

Your experiences:
You have told your watch that you are at a constant altitude. So any ambient pressure change measured by the watch should be used for calculating a new Barometric Pressure.

Your stair experiment shows that it did, but perhaps not as much as it should. A change of 1 mm Hg should be equivalent to an altitude change of 6-7 meter. So your 9-10 meter altitude change would have caused an ambient pressure change of around 1.5 mm Hg, and the change to the Barometric Pressure should be almost exactly the same. But it was only half of that. This can have two explanations:

• The ambient pressure sensor did not pick up the full change in ambient pressure.
  You may not have given it enough time, since the pressure measurement is somewhat averaged over time to prevent noise.
  This would be easy to verify by repeating the experiment, but this time you should look at the Ambient Pressure value, not the Barometric Pressure value.
  
• Your manual input of sea level pressure somehow screws up the calculation of Barometric Pressure. If you watch both the Ambient Pressure and the Barometric Pressure, and they do not change by the same amount, I would suspect this to be the reason.
  

Your observations over time in your apartment seems to verify that explanation #2 above is the reason.

Your readings from the Ambient Pressure widget points in the same direction.
In my watch, the third value in the widget is equal to the Barometric Pressure shown by the watch. If they are different in your watch, and you see the third value changing as much as it should while your Barometric Pressure reading doesn't change as much, I would again suspect that your manual input of sea level pressure was the reason.

However, you should be careful with the Ambient Pressure widget. As far as I can see in the discussion thread for this widget, the widget developer did not have access to all values in the watch while he wrote the widget, so he had to calculate some of them. The interface has recently changed so widgets can now access the necessary values directly, but I can't remember if this widget has already adapted to this. So there is a risk that the widget shows you a result of some indirect calculation.

Instead of using the Ambient Pressure widget, you could just make a custom data screen inside an activity app, for example the Walk app. This screen should contain these data fields:

• Elevation
  
• Ambient Pressure
  
• Barometric Pressure
  
• Temperature (not needed now, but maybe later).