Wet sanding using 1200 grit on my brown table has created a white/dust-like cover
I suspect the reason for the different outcomes is a result of using two different types of paint. The different properties and composition probably means their surface will respond differently to being sanded.
Many glossy surfaces will be dulled by sanding, even at 1200 grit. You were fortunate with the white table. Perhaps it doesn’t show the effect up as much because it is already white.
Sanding is an appropriate method to remove brush marks from a painted surface, but not as a final finish. It should be used as a preparation step before repainting.
Whatever the cause, you’ve not ruined your table, you just need to sand it correctly and repaint using a better technique.
The paint on your white table might now have an ‘open’ surface with micro abrasions from the 1200 grit. This could make it more susceptible to dirt or stains and it might need refinishing at some point.
For your brown table, you can prepare the surface ready for repainting and try again.

Dry sand with something like 120 to remove any remaining brush strokes and unevenness, followed by 180 to remove any scratches from the 120. Either way you need to rough up the surface a little because 1200 is too slick for the new paint to bond to properly. You could sand back to the original wood, or just stop when you get a nice even surface to repaint.
You could repaint using aerosol spray paint, which should give an even finish. Follow the instructions carefully especially about ambient temperature, distance, using several light coats and don’t put too much on at once so it runs.
You could try again with an oil based paint, but apply with a mini roller and a disposable short pile sleeve, followed by very light laying off with the tips of good quality brush to remove the roller stipples. Work quickly before the paint starts to skin over.
You could try again with a good quality brush. Work quickly and don’t overbrush the paint.

Personally I found that oil based gloss was easiest to paint to a flawless finish, followed by oil based satin and then water based needing the most skill.
Some tips here on getting a good finish: https://traditionalpainter.com/flawless-paint-finish