Why do you think Socrates himself used this method?

Was it mentioned that the intent is to have the other change their mind?

Should it have been?

Is expecting what it should be part of the Socratic method?

The point is that people do lower their level of confidence when they cannot prove their claims.

And this is exactly what the method accomplishes! Might as well call it verbal debugging.

What's the difference?

Grab a dictionary and look up "discussion", then look up "argument".

Discussion: consideration of a question in open and usually informal debate.

Argument: the act or process of arguing, reasoning, or discussing.

So, no difference?