Who defined the positive direction of the magnetic field?

[Hornbein]

This is irrelevant to your question but I can't help it. When geometric algebra is used for magnetic fields there are no arbitrary conventions. One gets a different sign in Maxwell's equations. I thought but never proved that a left hand rule would be the natural convention.

 

[Hornbein]

Faraday was the first to study electromagnetism so the answer may lie there. He was however not a mathematical guy. I seem to recall that the most popular theories were by Hertz and maybe Gauss. At the time, quaternions were in vogue. Maxwell used quaternions and was found later to be correct. So maybe the convention came from Hamilton. Later Heaviside's cross product took over. My guess is that by then the convention was already established.

Can quaternions be defined with a different sign convention? Beats me.

 

[Herman Trivilino]

the
end of a compass needle marked "north" is actually what is
called "south" on all other magnets.

Take a bar magnet and place it on a wooden platform so you can float it in a tub of water. You will find that when free to rotate the north end of the bar magnet points in the same direction as the north end of a compass needle. A compass needle is just a bar magnet mounted in a such a way that it's free to rotate.

 

[kuruman]

The magnetic field of the Earth can be modeled as a magnetized needle with its north magnetic pole (red) near the geographic South pole (see figure). Compass needles line up with the local field lines to minimize the potential energy. Thus, the north magnetic poles of the compass needles on or near the Earth's surface will point in the general direction of the Earth's geographic North.

 

[vanhees71]

It's drawn correctly but, from a physics point of view, it's incorrectly named in the above diagram: the pole in the north of the Earth is the magnetic south pole and vice versa. It's of course the usual terminology to call the south pole of the Earth (seen as a magnet) the "northern magnetic pole". That always leads to the confusion and the discussion we have right here.

The following picture from Wikipedia is even more explicit:

https://upload.wikimedia.org/wikipedia/commons/2/2b/Geomagnetisme.svg

It's much clearer, not to mention the Earth and compass needles first but just use magneto statics a la Ampere and define everything in terms of vectors (magnetic field, $\vec{B}$, and, if you want to deal with permanent magnets, magnetization).

 

[kuruman]

It's much clearer, not to mention the Earth and compass needles first but just use magneto statics a la Ampere and define everything in terms of vectors (magnetic field, B→, and, if you want to deal with permanent magnets, magnetization).

I think the confusion arises because of poor understanding of the dipolar field.

People know that two bar magnets line up [S-N] [S - N]. They would then assume that if we model the Earth as a bar magnet, we will also have [Earth] [Compass needle] ⇒ [S - N] [S - N] failing to realize that this happens only at the magnetic poles. It is natural to assume that the Earth's magnetic field, just like its gravitational field, has a non-zero divergence. I too had that misconception until I learned some physics.