A studio is a flat which has no separate bedroom, the bedroom is also the main living area.
So you may have your bed in one corner and your sofa and tv in another. Sometimes there is a seperate kitchen, but often the kitchen is in the main room.
Open plan living! A studio has it's own private bathroom. A bedsit would normally share kitchen/bath or both (gah!)
It's not as bad as it sounds, studios can have mezzanine areas or the beds can be in alcoves and it's fine, especially if you're on your own.
A double bedroom would have a double bed in it, a single bedroom a single bed.
Yes you can definitely find a place for �1K in zone 1 or 2, in fact you can have my flat in tower hill if you want! it's �867 a month, small one bed and the landlord always gives deposits back and doesn't ask for a credit check or references
If you're working in canary wharf have you considered Greenwich/Charlton/Deptford areas.
Greenwich is amazing with lots of little markets and nice shops and really safe. You could walk to canada water from Greenwich if you were so inclined or take the DLR.
You could also use spareroom if you're looking to bring your rents down, there's always lots of city people looking for flat shares, there's also flats on there to rent as a whole like studios etc.
If I could live anywhere it would be by the Cutty Sark in Greenwich, great boho lifestyle, cool pubs and restaurants and markets, so easy to get around.