For a long time babysitting was all that sitters could do. Then they branched out into house sitting and pet sitting. The latest in this line is elderly sitting where the sitter cares for an elderly person. A good elderly sitter will be there, not just to prevent crises and deal with them if they arise, but also to act as a companion, carry out errands as requested and be generally helpful.
The concept is still relatively new and so the term for it has not quite settled yet. Senior sitter is competing with elderly sitter, possibly because it is too easy to misconstrue the latter. A little alliteration always helps too.
Given that there is considerable enthusiasm for keeping people in their own homes as they grow older, the need for the elderly sitter will only increase.