So let me suppose that I measure no magnetic field, but I measure an electric field in the $x$-direction, and I measure you cruising past me at speed $v$ in the $y$-direction.
As you measure your fields, you will see two discrepancies with my measurements:
First, your electric field is a little stronger. It's hard to notice this because it does some interesting things to not violate conservation of energy; in particular the same strength increase will also be experienced by you as a time decrease in the amount of time that the electric field is acting upon you... also because the strengthening at low velocities is by the factor $1 + (v/c)^2/2,$ so even if you're going 100mph (= ~160 kph) you will only notice this if you can measure forces to accuracies of one part in 45 trillion (on the short scale, so a trillion is a million million, no "milliards").
Second, you will measure a magnetic field in the $z$ direction, and it will be approximately equal (barring the same sorts of high-accuracy equipment measured above) to the $v$ I measured times the $E$ I measured, divided by the speed of light squared. This is very weak for basically the same reason; $c$ is a big number and therefore $c^2$ is a big number. If you were traveling 100 mph past a $E$-field that was nearly strong enough to cause sparks through air (roughly 3 million volts per meter) then you would still only measure something like a billionth of a tesla. For comparison the magnetic field of the earth is about 30 millionths of a tesla, and there's a lab out there which specializes in high magnetic fields that can go all the way up to 100 tesla, with most other applications somewhere in the middle.
In other words, the "electric field" and "magnetic field" are not observer-independent as separate entities; if two observers are moving relative to each other then if one person says "the magnetic field did this, it's a magnetic effect," someone else might say "no, that's an electrical effect" and they might both be right in their respective coordinates. For this reason those two vector fields need to be combined into an "electromagnetic field tensor" which is observer-independent according to the coordinate transformations of special relativity.
Now if the electron is orbiting around the nucleus in the $\theta$-direction the $r$-directed electric field will appear a little bit like a $\phi$-directed magnetic field to the electron and can tend to interact with its spin. So that's where the spin-orbit couplings come from. If you froze the electron in a moment in time the proton would be moving relative to it and would therefore have a magnetic field circling around its direction of motion; this magnetic field is precisely the one that we're talking about the spin interacting with.