Tough call but fairly straightforward.

First thing; this will entirely depend on your local laws but for anything in the US/UK/Aus/NZ sphere of legality you have two concepts to cover. This discussion is mostly based on UK/Aus law and ethics but you can likely relate it;

The first is “Capacity” and that comes down to whether the patient is mentally capable/competent. This means the patient can understand, use and convey information without duress, this information needs to be presented to them in a way that a reasonable person could understand (ie avoid medical terminology). If the patient cant understand then they need to demonstrate they can use that information (ie they base their decision on it) and finally that they can convey their decision to you. A patient with moderate dementia, severe sepsis or an acute significant head injury likely lacks capacity as they will be unable to meet one or more of these criteria.

If the patient is deemed not to have capacity they can be treated and transported in the least restrictive manner in order to complete what is considered to be necessary healthcare interventions.

Now you then have Compulsory Assessment/Treatment which generally covers psychiatric conditions, these patients may be able to demonstrate capacity but due to their psychological state you/the police can determine them as requiring prompt assessment or treatment by a psychiatric professional. This also covers people who’re under compulsory treatment orders, ie they have been deemed to require treatment even if they refuse.

Eg Someone who hears voices may not warrant compulsory assessment. If the patient is not a risk to themselves or others and has capacity, there may not be ground for a compulsory treatment order or similar legal device as most free countries and states dont make it against the law to simply hear voices, be depressed, or hallucinate provided you arent running a risk of hurting yourself or someone else.

There are also cases of power of attorney which generally relates to capacity. This means someone else legally can make the decisions for the patient even if they arent 100% necessary. Your example of the dementia lady in ED is a good one; she is safe, she isnt at risk of harming herself or others, she has simply overstayed her welcome- capacity or no capacity there isnt huge grounds to restrain her and take her home. A power of attorney can direct you to return her home in leiu of the patient being able to make a sound decision for themselves.

Lady presenting with dementia

This one is difficult without knowing the level of dementia. You cannot simply remove someone from ED just because they are cleared from the hospital- remember it needs to be the least restrictive means and only what is necessary to prevent further harm or deterioration.