"Agile is a methodology used to develop your work processes"

No, more often it is a noun that is packaged and sold by consultants rather than the adjective describing an outcome as it should be.

"...it is not a process itself."

It shouldn't be, but when you push for focusing on the process...

"Agile can be applied to any business,it is not a software development technique."

The title is, "The Manifesto for Agile Software Development". This conversation is also focused on software. All other contexts are irelevant.

"What you described is very important too - keeping the code lean and QA high, but it should be the result of a well applied Agile."

No, it's the outcome of focusing on smaller batches delivered more frequently with fewer defects. That does not flow from 2-week sprints with long delays between improvement cycles and big-bang improvement items. That comes from solving the people, process, and tooling challenges of continuous delivery. When you finally achieve a CD workflow, you'll find that many "truths" pushed by the Agile Industrial Complex are contextual to the very slow feedback loops and big batches of 20 year old methodologies. We've moved on. If you claim ti be "agile" in 2022 and cannot release daily, you're deceiving yourself.