I agree with Louise, don't rely on the stairlift company for info about whether the stairlift is suitable, they just want a sale. I am sure it looks safe to you, but people with dementia struggle to learn new things. A year ago when she was still living at home, my mother couldn't even remember to leave a tracking monitor on her belt, she removed it as soon as the carer left, because to her it was a random foreign object. If she saw a stairlift I think she would have just walked past it as she'd have had no idea what it was for.
Re how you approach the subject of dementia with your mum, there is no need to do that. Call it memory problems, or whatever works best for her. I would think though you would need a diagnosis for her to be accepted into a dementia care home. The GP can give a mini memory test and then refer to a memory clinic. But you're jumping ahead a bit, at the moment it sounds as if she could cope with carers coming into her home, and you don't need a diagnosis for that, although it would be helpful to start the process towards getting one.
My mother is not in a nursing home, she is in a dementia care home. It is true that some people will need a 'dementia nursing care home' to deal with both dementia and medical needs, but by no means all. Nursing care is care which only a medical professional can carry out, but many dementia patients only require 'personal care' i.e. help with dressing, washing, eating, mobility, continence. My mother's care home deals with all this, including residents who are in wheelchairs, are bedbound, and/or need hoisting to move them, and they aim to care for them until end of life. Any care home (or care agency) you approach will want to meet your mother and assess her themselves to ensure they can meet her needs.
I know it is a stressful difficult time and we all learn as we go along, but you will find a way through to get her the care she needs.