Is it possible to use the word "patient" for the people who are cared for by caregivers or the people who are in need of nursing care? If not, what do you call those people?
"Patient" is derived ultimately from "patior," a Latin verb meaning to "suffer." Through a convoluted process, it has come to mean a person who is being treated for illness or injury. As such, it is certainly used for people who are being cared for by a nurse.
It is not typically used for someone young and healthy. No one says that a child is a patient of its nanny.
It can be ambiguous when used with respect to the old who are merely frail. "Patient" usually implies disease or injury, and old age is not typically considered a disease. I at least would find it odd to hear a mentally competent, physically healthy old person called a "patient" in most contexts. But context can never be ignored. In the context of a nursing home that cares both for those who are merely frail and for those with serious physical or mental deficiencies, it may be idiomatic to refer to both classes as patients. Personally, I would tend to reserve "patient" for those who are being cared for due to illness or injury rather than for old age, but I will not say that that preference represents general usage.
Patient can be used to describe somebody who is not actually ill:
1 b : the recipient of any of various personal services
2 : one that is acted upon
// are agents as well as patients and observers in the world
— C. H. Whiteley
Another common word is client:
2 a : a person who engages the professional advice or services of another
// a lawyer's clients
// a personal trainer … enjoyed the challenges of helping clients buff up their bodies.
— S. K. Parks
2 b : CUSTOMER
// hotel clients
// a restaurant's clients
2 c : a person served by or utilizing the services of a social agency
// a welfare client
Which word to use is a matter of context.
In terms of what's more common, you can refer to Google Ngram Viewer results for various situations, which indicates the word more often used in print.
Nursing home patient is much more common than nursing home client:
On the other hand, massage patient is used so infrequently that it doesn't even register at Google Ngram Viewer, which shows only massage client:
Finally, welfare client has an edge over welfare patient: