куртка is a coat too, same as in Russian. a short one. Because you could wear a куртка in winter. especially if it's lined with down. You wouldn't dream of wearing a jacket when it's significantly below zero.
Yup. Would like coat accepted (because I keep getting it wrong! Can't never remember to use 'jacket' :-O )
Should be pronounced мЕне, not менЕ because it is preceded by a preposition