I learned that the kind of stove and fuel makes a big difference.
My old stove, a Trangia, preferably burns denatured alcohol (aka methylated spirits). It's a pretty small flame in regular use, and no flare upon lighting it. The flame is enclosed very well, but if the burner is tossed over, the alcohol can easily escape. Inside a tent, that would lead to a life threatening situation, but the design of the stove makes it very hard to accidentally topple it. As long as my tent is able to withstand the storm, I can cook inside the closed vestibule, or, for added warmth, inside my tent - but the latter only when I'm alone.
Now, the main problem when cooking with alcohol is that it's slow. No problem for simmering, but it takes a while to get your water to a boil. Normally, I can live with that, because there's always enough to do during cooking - I'm talking about real cooking here, not just pooring water into freeze dried meals. However, recently I learned that at high altitude, where cooking takes longer anyway, it was just not workable anymore. And the final drop was that melting snow and making tea for breakfast took hours. So I started looking for another stove.
Climbing with different people gave me the opportunity to see different stoves in action. I had already read about the Jet Boil, which uses gas canisters. I didn't really trust it inside my tent, until one of my climbing partners brought one along and I saw for myself that there was no flare, and a relatively benign flame. Since the flame is not protected, and the burner can easily be toppled, I didn't want to use it inside my tent, but I used it in the closed vestibule in really crappy weather. It's much faster than alcohol, good enough for melting snow at high altitude, but I decided it wasn't for me, because gas isn't easily available all around the world. Besides, I had seen other stoves.
The one I decided to buy was an MSR Dragonfly, which preferably burns white gas. As for speed, it's just as fast as a Jet Boil for boiling water, and for real cooking it's in a league of it's own. However, it can give a big flare upon lighting, and the flame is wide open, making it very dangerous inside a tent. In bad weather, I open up the vestibule as much as the weather allows, then light the stove under a bit of open sky. The walls of the vestibule act as a wind break. There is no shelter against snowfall, but, more importantly, any flare will be harmless now. After lighting it, I sometimes move the stove a bit closer to the tent itself, but keep it in the vestibule. I don't know if this procedure will still work in a really big snow storm, haven't tried that yet. However, I guess that in that case I want to be in a snow cave instead of a tent anyway, and then there is no danger of burning my shelter. And I haven't figured out yet what to do when it's warmer and raining heavily.
So, to sum up my experiences:
1) Alcohol: relatively safe but quite slow, especially at high altitude. I've sometimes used it inside my tent.
2) Gas canisters: bit more dangerous and fast enough, even at high altitude. Never inside the tent, but usable in closed vestibule.
3) White gas is very dangerous and can give a flare, but (with the right stove) fast, even at high altitude. Only in open vestibule, and I don't yet know how to handle heavy rain.