What is the difference between the usage of "doing art" and "making art"?
As you noticed, sometimes they're used interchangeably. However, there could be a bit of distinction that I will try to tease out. I'm not certain because I'm not a visual artist.
I think "I do art" means that my primary activity is art, with the focus principally on the creative process. I think, therefore I am --> I am alive, therefore I do stuff with ... clay, paint, found objects, chalk, charcoal, etc.
I think "I make art" focuses a bit more on the output.
However, musicians use "make" somewhat differently. They don't say they "do music." For them, the prosaic verb is "play" music. The lyrical verb that musicians like to use, to express how sublime it is to create music, is "make" music. When they talk about making music, you might almost get the feeling they're talking about an activity akin to making love. "Making music" is romantic and magical. "Playing music" is a bit more workmanlike.
I suspect visual artists' use of "I do art" is closer to "Let's make music together"; while "I make art" is closer to "I play music" -- a more literal, technical description.