Memory care vs. nursing home vs. assisted living?

Memory Care, usually there is staff specifically trained in various types of Dementia. A resident may come into the facility with some functions and awareness but they wander. Most Memory Care facilities are locked units VS Assisted Living that may not be a locked unit.
Assisted Living residents can still do some ADL (activities of daily living) but may need help with some. This would be people needing more help than independent living. (Usually no skilled Nursing other than medications)
Nursing Home this would be skilled nursing. In many areas once a person needs more than 2 person transfer meaning the need for a Sit to Stand or a Hoyer to transfer they need Skilled Nursing or Nursing Home care.
Some Memory Care will permit the use of equipment if the resident entered the facility not requiring such care but has declined to need that level of care.
Each facility will have different requirements. And I think states may also have different guidelines .

I'm no expert, but, in my state, a Memory Care unit or facility is considered Assisted Living, but, they qualify as a Specialized care unit. Meaning, they do much more for a resident than a regular AL. Most residents need doctor to say they need this kind of specialized care. The MC may need to do everything for the resident, from changing depends, bathing, dressing, even hand feeding. They can't handle skilled nursing though.

In my state, a regular AL can do A LOT for a resident, but, they have some requirements that they can't disregard. Such as they can't keep a resident who requires more than a one person transfer assist AND they can't be double incontinent. Also, a regular AL could not manage the care of a resident who wandered. The goal for the MCs in my area is that they stay there for the remainder of their care, however, they cannot manage the care of someone who needs skilled nursing care.

Someone who needs skilled nursing care, like who has tubes, would need a nursing home. I hope this helps. You'll probably get more responses too.

Your sister-in-law is in denial about her moms condition.

I wouldn't worry what it's called as long as your MIL is getting the care she needs.