I think the replies above are great - and my only reason for posting is to give my experiences of moving into storm photography a number of years back - I've always taken video, but wanted my own camera for taking stills.
I decided to not go down the DSLR route, as a) I wasn't really sure how much I would use it, and b) I didn't want to carry multiple lenses around, especially considering a). I've still got the camera - and have not moved into DSLR yet - mainly due to being happy with it, and it doing what I need from it.
I bought a Sony RX10, back in 2014 - it's a fixed lens camera, and techinically classed as a 'compact' - but it's a similar size to many DSLR cameras. It has a 24-200m f/2.8 lens, which gives great results - with f/2.8 at 200mm, you can get good light collection even at a decent zoom.
The RX10 has moved on to version 4 now (I've not moved on, mainly due to cost!) - it is packed with a lot of tech, and now has a 24-600mm lens, F/2 - 4.4. For an all-in-one camera, I think this really is pretty high!
Downsides? The fixed lens means you are 'stuck' with those ranges and f numbers, which is an issue if you want maximum flexibility. Also, the sensor size is 1 inch - so, no full frame...for many applications this is OK, and I find night photography is pretty good. However, at higher ISO it does get pretty grainy, when a full-frame would still be sharper/less noisy.
Below are just a couple of storm shots, which have been tweaked in Lightroom to improve contrast, etc - it shoots in RAW, which is very useful.