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Care-giver is used in the UK, but usually for someone who is not the primary carer, or for whom care-giving is not the primary job. So for instance a carer might take their client to a location where additional assistance is needed; a porter or assistant would help, and would temporarily become a "care-giver" whilst the client is still in the primary care of their carer.
"Care-giver" is a more generic term. There are plenty of people who do not require a carer (they are mostly independent) but occasionally need help in a specific task. They might be offended by that care-giver being referred to as their "carer" since that implies they are dependent upon that person.
Isn't it weird that A is a care-giver AND a caretaker!?
A: Care-giver (gives care)
B: Caretaker (says "thanks you" and takes the care given by A)
Also, why does one have a hyphen while the other doesn't?
None of this makes sense