Attitudes towards learning seem quite, quite different between Korea and the western world. I'm thinking maybe a look at the different definitions of the terms might help clarify the gaps.
1) give one's attention to acquring knowledge (of a subject) - 지식을 습득하기 위해 주의하다
(eg. study economics at university, study music in Italy, study in private)
2) examine attentively - 체계적으로 연구하다 (잘 주의해 보다)
(eg. study social change in the region, study a picture)
1) gain knowledge or skill in - 지식이나 기술을 얻다 (익히다, 습득하다)
(eg. learn to cope with a busy schedule, learn to tell right from wrong, learn a language, learn to trust, learn to knit)
2) become aware of - 눈치채다, 알아차리다
(eg. learn a rule of thumb, learn where the limits of tolerance lie, learn from history)
3) memorize - 암기하다
(eg. learn this for the test tomorrow)
1) 남에게서 가르침을 받다. - receive teaching from others (teachers/books/etc)
(eg. take piano lessons, take swimming lessons, take driving lessons)
2) 남의 행동·태도 따위를 본받아 그대로 따르다. - imitate others' actions/attitude/etc. Just follow.
(eg. A growing person watches his or her elders and imitates them.)
3) 어떤 습관이나 습성 따위를 붙이다. - take up a habit or behaviour, etc.
(eg. take up smoking, take up drinking)
4) 지식·기술 따위를 익히고자 힘쓰다. - Make an effort/work hard to pick up knowledge, skills, etc.
공부-하다 (gong-bu ha-da)
학문과 기술 등을 배우고 익히다. - (do the above and) acquire learning and skills, etc
1) 익게 하다. - ripen; get used to, accustom oneself to
2) 익숙하게 하다. - familiarise oneself, develop (a skill), learn/acquire (a language)
I would suggest that the most common of these three Korean words, gong-bu ha-da (공부하다), can be expressed along the lines of "to learn through study", stressing that to learn you have to put in a lot of effort. (Compare this to 'study', stressing attention though not necessarily effort.) It also seems to suggest a reliance on others, rather than 'learning for oneself'. (Compare this to 'learn', which suggests gaining though not necessarily implying received knowledge.)
'I-ki-da' seems to be closest to 'learn' or 'acquire'.
'Gong-bu ha-da' seems to be 'learn (sth) through study'.
'Bae-u-da' seems to be 'learn (sth) from others' and (4) 'work hard at (sth)'.