As a woman and a nurse, I've never had the experience of being mistaken for an MD. Just one of the stranger thing that happens when people expect one thing, see another.
As an NP I have had people insist on calling me doctor, doc... I tell them, over and over, that I'm an NP. Most end up saying, I know, but I'm calling you Doc.
As long as they know and I can't get them to call me anything else, I go with that.
As to a confused patient - say it, if they don't get it, I think it's too confusing for them if you go on. A sick, confused elderly woman sees male, white, MD. As long as you don' t take on the role, your OK. Just have to be careful, make sure she'll let you help her with that "nursing" stuff like bathing, go to BR, etc. Thinking you a doctor, she won't think to ask. I'd just recommend that you try to anticipate.
There are advantages to being treated as a doctor - patients tend to be nicer, treat you with more respect. They may give you information they would not give you if they thought you were a nurse. The depressed and suicidal stuff - don't know if she would have brought it up. Also, patient feels special, cared for.
Disadvantages: for me - I'm not a doctor, want to be a nurse and sometimes wish people could respect that without adding label "doctor." I'd also rather have people feel special and cared for when it's a nurse doing it.
Just my 2 cents. You're doing great already by asking others how to handle. It's hard to figure on your own.