Why some Author's write that SCRUM is a methodology?

SCRUM as a methodology,

Scrum is a process framework, Agile is a methodology. But since many have a hard time distinguishing between the two, and these concepts are incorrectly mashed together as one, this unclearity happens

Is a Scrum Master the same as an Agile Coach?

It depends on the organization. In some, Agile coaches are Scrum Masters who are in a management position and therefore are more or less a Scrum Master with higher pay ;)

Following the Scrum Guide, what an SM is is clearly stated. Agile coaches are not. No one can say what an Agile Coach is, it depends on context and organization. In my experience, Agile coaches tend not to focus on Scrum and Agile on team level, but more on organizational level. Agile transformation in the tiers above team level, up to C level.

In my opinion, Agile is a mindset manifested through the 4 values and by the practice of the 12 principles. I wouldn't really call it a methodology.

A Scrum Master is a role within Scrum, has a broad skillset, has knowledge of other complementary frameworks, methods and approaches, serves the Development Team, the Product Owner and the Organization. In that sense, it could be said that a Scrum Master is an Agile Coach too.

Frankly many times when some one says Scrum methodology they mean Scrum BASED methodology.

Pay attention to this phrase: Brick is a construction material. It is a material that is used to construct structures and buildings.

Similarly(compare the bold words): Scrum is a process framework. The framework can be tailored and used to build our own process or methodology, but it is in the end a FRAMEWORK , just like brick is in the end a material. What you want to construct with it is up to you.

Coming to your second question. An Agile Coach is basically an experienced Scrum Master. As a scrum master you coach and train your team and product owner on a daily basis in order to champion scrum and make sure that Scrum is adhered to and followed. With experience you will be able to coach other scrum master, at which point you can be can be considered an Agile Coach. Agile Coach is not a scrum term, but considering it's usage among other agile methodologies, an agile coach focuses more on the enterprise and organizational transformation while Scrum master is more focused on the team and Product owner.

If (organisation applies Scrum)

       Then (Scrum Master)



Else (Agile Coach)