I do weather photography on a personal level so I have rather good experience with it.
You didn't say specifically what you wanted to shoot or what you felt you needed the extra wideness for. Was this for clouds, lighting...?
CLOUDS:
For clouds you need something a bit wider then what you currently have. I would recommend sticking with something in the range of 14mm+. If you go wider then that you are going to start to get some distortion at the edges, which only gets worse as you go down in focal length. This can however be fixed in post to some extent. If, however, you like the fisheye look, then this does not matter.
LIGHTING:
For lighting wider is not better. Lighting should be in the distance when you are shooting so you should need something in the range of 20mm+ to capture it. If you need something wider then YOU ARE TOO CLOSE!! Lighting on a wide angle lens looks small and lacks detail. The only way to get lighting to look good on a 10mm lens is to have it within a mile or so of where you are which is extremely dangerous and asinine.
LOW-LIGHT: You don't need a fast lens for weather photography, even at night. Weather photography is simply landscape photography and this is usually done at higher f-stops. All my weather-related photographs are shot at f/8+ (even at night) so the extra stops of light in a f/2.x lens won't help you unless you decide to use it for something else as well.
Recommendation:
My honest recommendation for all-around weather photography would be the EF-S 15-85mm f/3.5-5.6 IS USM. Seeing that the current lens you have is slightly less wide then this, except for the overall IQ, there would be no reason to upgrade to this lens. That being said, the 8-16 would be the most versatile since between the two you have 8-85mm at your fingertips. The 8mm would be good for clouds, rainbows, etc. while the 17-50mm range of the other glass would be good for lighting. You may also want to look into the Tonika 11-16 which is a very good lens (if you can find it - they have been out of stock on and off for years). Reviews on the Tonika have said that it is very sharp with very good IQ (even beating out some of Nikon's best glass).
All this being said, I still do have to plug the EF-S 15-85mm f/3.5-5.6 IS USM which honestly is my favorite wide-angle zoom even among the L glass. If you can afford the upgrade, I would replace the 17-85 with this.
I skimmed over my online uploads real fast. Here are some random shots I have taken with that lens:
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