słownik nie mówi o 3, różnych znaczeniach ?

Tak. Słownik Diki zawiera objaśnienia tych słówek. Jednak jeżeli sprawia Ci trudność wyłapania różnicy między nimi, to mój skrót powinien Ci pomóc rozwiązać Twoje wątpliwości:

1. vain - próżny, zadufany w sobie
2. conceited - zarozumiały, przemądrzały
3. boast - chwalić się, szczycić się (jedno ze znaczeń tego słowa jako rzeczownik brzmi: duma)

Myślę, że teraz masz to przekazane prosto, więc teraz polecam metodę "kopiuj -> wklej -> formatuj -> drukuj -> na ścianę i lukaj na to od czasu do czasu :)

Hejka

----------------------------- VAIN -----------------------------
vain (adjective) = NOT SUCCESSFUL, USELESS, CONCEITED
unsuccessful or useless; of no value
(*) The doctors gave him more powerful drugs in the vain hope that he might recover.
(*) (formal) It was vain to pretend to himself that he was not disappointed.

vain (adjective) = SELFISH
too interested in your own appearance or achievements
(*) He was very vain about his hair and his clothes.
--------------------------- CONCEITED ---------------------------
conceited (adjective) (disapproving)
too proud of yourself and your actions and abilities
(*) Without wishing to sound conceited, I am the best salesperson in the company.
--------------------------- BOASTFUL ---------------------------
boastful (adjective) (disapproving)
praising yourself and what you have done
(*) I tried to emphasize my good points without sounding boastful.
----------------------------------------------------------------
Describing arrogant and over-confident people or behaviour:

arrogant
someone who is arrogant thinks they are better or more important than other people and behaves in a way that is rude and too confident

proud
used about someone who feels so proud that they think they are better than other people

conceited
someone who is conceited behaves in a way that shows they think they are very intelligent, skilful, or attractive. This word shows that you do not like people like this.

immodest
someone who is immodest thinks they are very good or clever and likes to talk about their achievements

superior
someone who is superior behaves in a proud way, as if they think they are better or more important than other people

vain
someone who is vain is very proud and thinks they are very attractive or special

egocentric
behaving as if you are more important than other people, and need not care about them

patronizing
behaving or speaking in a way that shows you think you are more intelligent or important than someone

big-headed
behaving in a way that shows that you think you are very important or intelligent

loud
used for describing a person who talks in a loud and confident way that annoys other people

Pozdrówka

Myślę, że moi poprzednicy wyjaśnili Pani różnicę między podanymi słowami i nie muszę się już wtrącać do dyskusji:)