I believe the name of this branch of philosophy is aesthetics, which you actually already mentioned in your post.

You may be talking more about art therapy than a form of philosophy in referring to art to express oneself but it could just come under the category of creative art. Aesthetics is probably a lot more complex because it is a field of philosophy in its own right. When you speak about a person drawing circles and geometrical shapes, I am of the view that it is best not to try to categorise it with a specific meaning outrightly but to ask the person to say what meaning this has, because this would depend on the significance for the person making the art.

If you look to your left - third box down, 7th entry down - philosophy of art!

What about Stoicism?
Is there a philosophical theory or framework that is more specific than 'aesthetics' which relates to what I describe in my initial post?

express oneself by drawing art, to externalise and project emotion through art, to have artistic understanding, to have a sense of beauty / aesthetics, to be logical? — User34x

You're mixing up so many things here that I can tell you there is nothing else closer than aesthetics.
Breaking-up your points:
"externalise and project emotion through art" - maybe better to look at Art-therapy
"to have artistic understanding" - you need to study history of art or as well call in europe: Beaux-arts.
"sense of beauty / aesthetics" - aesthetics
"to be logical" - epistemology