"I left smoking", "I quit smoking", "I gave up smoking", "I stopped smoking" are these same?

"Quit" implies it was an intentional action.

"Give up" also implies it was an intentional action, but suggests that one would have liked to continue doing it.

"Stop" is neutral regarding desire and intention.

I think they all are taken to generally imply permanence. Hence "Quitting smoking is easy, I've done it half a dozen times" is a joke (violating one's expectations, he hasn't really 'quit' at all).

"I'm not smoking" doesn't imply any permanence. To emphasize permanence you could say "I've stopped smoking for good."

"I left bringing my bag" seems incorrect to me. I would parse it as the oddly phrased "I left, bringing my bag". If someone has "left acting" or "left boxing" that makes sense, because "acting" and "boxing" are nouns (occupations) but "bringing" only exists as a verb form, and can't be the object of 'leave'. By that argument, "I left smoking" is also incorrect.

Verb + -ing form strings have various structures.

[A] as analysed by Collins

[From the CoBuild works:]

V -ing constructions:

The verb is followed by an '-ing' form.

This pattern has three structures:

Structure I: Verbs in phase (i.e. complex verb groups) She started walking.

Structure II: Verb with Object He liked dancing with her.

Structure III: Verb with Adjunct They ended up fighting.

There is also the even less cohesive structure, which uses the ing-form or participial construction as an adjunct:

He went, singing. / He went, singing at the top of his voice.

This is not the same structure as the phase structure

He went singing / shopping / dancing / bowling / fishing.

Sometimes, these strings are hard to analyse.

Under the trees, he stood watching the fight. [probably a phase structure]

Under the trees he stood, watching the fight. [definitely not catenative]

He left singing. [left the profession]

He left, singing. [he seemed happy as he left]