Naming a role people were already doing
On the language used for the role, and how the available words shape what is expected of it.
The people doing this work had been doing it for a very long time before there was a common word for them. They were daughters, sons, wives, husbands, neighbours. What they were doing was covered by the relationship, and the relationship was assumed to include it. There was no separate term because there was thought to be nothing separate to describe.
The arrival of a general word for the role is recent, and it did something more than provide convenient shorthand. Vocabulary determines what can be counted, claimed, discussed and compared, and a role with no word for it exists only inside the households where it happens.
What a name makes possible
Before the word, a person could describe what they did but not what they were. This sounds like a fine distinction and it is a practical one. A description belongs to your particular circumstances. A category belongs to a group. The first invites sympathy for your situation; the second makes it possible to ask whether other people's situations are the same, which is the first step in almost every change of common understanding.
It also made the activity visible to anyone counting. Whatever is not named is not enumerated, and whatever is not enumerated is absent from every discussion in which numbers are the currency.
The trouble with the available words
The words in general use all carry difficulties. Terms built from the verb to care suggest an emotional disposition rather than a set of tasks, which quietly implies that someone doing the work without much warmth is doing it wrong, and that someone with plenty of warmth needs no practical support.
Terms borrowed from paid occupations have the reverse problem. They describe the tasks accurately and imply a training, a shift pattern and an ending, none of which apply to a person doing this in their own home for a relative, indefinitely, without having applied for the position.
Why many people decline the word
It is very common for people to reject the term when it is offered to them. They will say that they are not a carer, they are simply a wife, or a son, and that what they do is what anyone would do. This is usually said firmly and it is worth taking seriously rather than treating as denial.
The refusal is generally a defence of the relationship. To accept the occupational word can feel like agreeing that the relative has become a task, and that the relationship has been replaced by a function. That is a real loss to guard against, and the language does genuinely threaten it.
The awkward consequence is that the vocabulary which makes the role visible in general is the same vocabulary that many individuals find intolerable in particular, which is a large part of why so many people doing this work are not counted among those doing it.
What the words do to expectations
Language also shapes what is thought reasonable to ask for. Describe the role in terms of devotion and support becomes a kindness offered to a deserving person, which can be declined for reasons of modesty and withdrawn for reasons of budget. Describe it in terms of work, hours and coverage and support becomes a requirement of a functioning arrangement.
Neither description is complete. What people actually do contains both a relationship and a great deal of unpaid labour, and any single word will suppress one of them.
A practical position
There is no need to settle this. It is enough to notice which word is being used and what it is doing. Within a family, the language of relationship is usually the right one, because it is true and because it protects what matters. When arranging things, dividing work, or asking for something specific, the language of tasks and hours is more useful, because tasks and hours are what can be allocated.
Using different words in different rooms is not inconsistency. It is a reasonable response to a role that genuinely is two things at once, and has never been well served by having to choose.