When used in the present tense, presently is indeed a synonym of currently. When used in the future tense—e.g., “I will be leaving presently”—it means “shortly,

When used in the present tense, presently is indeed a synonym of currently. When used in the future tense—e.g., “I will be leaving presently”—it means “shortly,” since “currently” isn’t compatible with the future tense.