잘랐습니다 sounds more natural for women in everyday Korean
깎았습니다 would usually apply to only men. Perhaps in rare cases of 비구니 (Bikkhunis / Buddhist nuns) cutting their own hair?
Do most men cut their own hair?
머리를 자르다 = to cut one's hair
머리을 깎다 = to trim one's hair
깎다 indicates shorter length being cut off; 자르다, longer length.
Why 깎았습니다 instead of 잘랐습니다 ? The hints on hover used 자르다.
I thought 'hair' would be 'morikaragi'?
Literally, 머리카락 means hair (/head of hair) and 머리 stands for head.
But in contemporary usage, 머리 is used to mean both.
머리 also is used to mean "brain" as in brainy/intelligent.
털 = body hair; also, animal fur