I have never heard any person actually use the word "ajar" -- unless you count my brother's friend's GM, which had a recording with a southern accent -- "A door is a-jar." I always thought it was a strange expression. But I think it means entrouverte, only slightly open, rather than half-open, as the Aussies seem to indicate.
BrE uses "ajar" quite frequently. I agree with you, though, that it means "slightly open" and not "half-open".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.