The sentences ‘I don't care’ and ‘I don't care one way or the other’, used so often as near-synonyms, are just occasionally completely different.
This has probably been obvious to everyone else for ages, but I've only just noticed it. It's perfectly possible to care in general about a situation, so that the pain it causes becomes your own pain and so that you wish to lend your strength to help achieve the most favourable outcome, without having a strong opinion one way or the other about which of the mutually exclusive possible outcomes would actually be the most favourable.
(I'm having those half-asleep long thoughts again, I'm afraid. When I'm sleeping in until nearly noon and lazing around all day, they don't seem to be restricted to the early mornings any more.)