Different methods to define an azimuth? what are they called?

There are three common to the US and most countries.

Magnetic North
True North (this is the point which the earth rotates about and is not the same as the magnetic pole)
Grid North (militaries use a grid system which is not lined up with either).

Since solar design is concerned with solar and solar is concerned with the position of the Sun, the one that matters is True North. The axis of rotations is related to the position of the sun. True North is also the north you will find by any of the many solar and stellar direction finding methods that use the sun and time, or stars etc.

SMA web page uses their own funky system. system which is not an azimuth despite what they call it.

if you go on your smart phone you can have the compass give you true north instead of magnetic north.

Good question.

I'm not sure what the SMA folks are using as I've not seen it, nor do I know what various conventions may be called, but the SMA system may be the one in use by some authors ( see Duffie & Beckman for example ) that defines true north as 0 deg., east as -90 deg., west as +90 deg. and west as 180 deg. In the past, that convention made some solar angular calculations a bit easier, particularly the hour angles.

I'm guessing here, but that convention may have come about as a result of an attempt to make things easier before calculators and later computers made converting, say 195 deg. in the "correct" system to a more manageable 15 degrees that could be more easily be found in tabular format in a handbook. Conversion from one convention to the other for machine calcs is not difficult once the system in use is understood.

When programming such things in the past (anyone remember FORTRAN IV ?), I've used the system that the algorithm I'm using and following has, and convert the angular references to the more conventional "zero degrees == north" convention in the output. I made fewer errors that way.

Things are a lot easier now with canned software - see the NREL stuff - folks would have killed for that computational strength in the '70's, but the learning experience and the familiarity that comes with it have mostly been lost.

I threw out my Fortran book in this weekends major cleanup. Bruce Roe

LOL. I think I used mine to level a desk for years after I graduated from school.

What I know is that azimuth determines the direction of the celestial body. It is the angel between sun, moon and the north which is measured clockwise.

That is but one use of the word. Wikipedia is your friend. Check it out.