I agree that 'art' means 'painting' unless otherwise specified. An art exhibit would show only painings and maybe drawings, most likely not sculpture or other art.
Painter to means what the English call 'painter-decorator'.
My mother painted. She was an artist. If anyone called her a painter she would be cross and point out that 'painter' suggests you are decorating the house. Artist is far more commonly used when refering to someone who paints pictures.