“The common-sense understanding of the word ‘method’ may be something like: ‘a set-up of presumptions and techniques used systematically to arrive at a certain result’… A method, understood as a procedure or a process, should be something going on between theory and practice.”

Let’s say, now, that ‘method’ is a way… a human act of making out a direction from what is perceived (the sense-world) and conceived (thought).

This is rather a broad description.
A special case, with a more narrow, or precise, definition, would be ‘scientific method’. ‘Scientific method’ – and the science produced by it – may be seen as the core of modernity.
In the post-modernity we currently share, some people would have science devaluated to ‘just another story’; a contextual truth told by just another group of people (male white Western scientists?) to fit their own agenda.
The funny thing about this attempted devaluation is that it frequently seeks to legitimate itself exactly by the (ab)use of scientific terms – preferrably fetched from the most prestigious fields of mathematics and physics.
In this aspect, mathematicians and physicists Alan Sokal and Jean Bricmont pointed out some French philosopher queens and kings to be very naked. This was done already back in the nineties; I cannot find that kind of philosophy better clothed today.

Logic thinking, scientific method and quantitative research springs from a millennia-long, careful cultivation of thought – not only in Europe.
On the other hand, science cannot (and does rarely) claim to interpret our life-worlds fully. In everyday life, we alternatively employ logic thinking and the language of art and myth.
It shouldn’t be too provocative to say that logos and mythos are both generic human modes of thinking; that they both tell us beautiful and challenging truths; and that they ought not be confused.

The initial quotation is from a text of mine called to care in a peculiar way; see page in English.
For Bricmont’s and Sokal’s elucidating review of some post-modern philosophers’ methods, see Impostures intellectuelles (French version) or Fashionable nonsense. Post-modern intellectuals’ abuse of science (English version); Bricmont and Sokal 1997/98.