Canon PowerShot G1 X
If you're willing to stretch your budget a bit, I'd suggest this.
-very good quality photos
Definitely
-future proof (atleast for a while)
It's the only compact out there right now with as large a sensor, which gives it better low-light high-ISO performance than most other compact cameras
-it would be nice if I could buy an underwater case for it should I need to(not a major point but will be nice)
Waterproof Case WP-DC44
-rechargeable battery that lasts
Ayup. Supplied Li-Ion.
-budget friendly(willing to pay more if its an excellent camera)
You'll grow with it rather than having to keep buying a new camera to grow with you. The ring around the lens can be used to adjust focus, it has RAW file support, it allows for in-camera HDR if you want that, it has an articulating screen, it has an optical viewfinder and has all the manual controls you could ever want - even more than consumer-grade DSLRs do. It also supports an external hotshoe flash.
You're basically just looking at a camera that's a bit fatter - it's not necessarily going to fit too nicely into your pocket, though you can get a belt-attachable pouch for it, or just a small shoulder camera bag like
this one.