A bedsit is like a studio flat. It is a single roomed accocmodation containing bed, cooking and living facilities:
I have never heard of the word "bedsitter", but if you said "bedsitter" to me, I would take the meaning as someone who is staying at my bedsit and looking after it for me, whilst I was not there.
Similar to the word "Housesitter"
‘Her presence means he has a house-sitter during his many forays abroad, and when he is at home he enjoys her company and finds security in it.’
I may be wrong, but that is how I understand the word. I'm British by the way!