The terms used vary with different areas, but I'll have a go at clarifying.
Residential - infirm, no nursing requirement, not suffering with dementia.
Residential with nursing - nursing required.
Residential dementia - suffering with dementia, secure building, dementia trained carers.
Residential with EMI - suffering with dementia, secure building, dementia trained carers and nurses.
EMI stands for elderly mentally infirm is an old term that does not appear in current CQC documentation.
I've never heard of EMI Zero.
There are also statements about staffing levels and the phrase to be used where a person suffers with advanced dementia, does not have any challenging behaviour, and requires nursing.
In the final analysis there should be a care plan in place that can be shown to prospective care homes who can then decide whether they wish to do an assessment as to whether or not they will accept the person. Worrying about how the home is described is in some respects a waste of time.