it's a compicated topic!
Viskovatov says the shade of the coats for the regiment should be uniform but the exact colour is down to what's available.
Even if all are in grey, I think the officers would have a nicer coat (& probably of a diffrent hue).
especially in 1812 the Russians had shortcommings with uniforms. The new recruits for the infantry fought in the grey training uniform at the French retreat. Take a look at Osprey Russian Grenadier for more details.
So when the simple infantry uniform was difficult to get what about a flamboyant hussar uniform? The Russians had the same problems as the allies with the cold and winter clothing.
I don't know any contemporary painting showing Russian hussars in greatcoat. So I fear we have to trust the secondary sources. Or does anybody know a primary source?