You are mixing together several disciplines at several different levels, so it is, indeed, confusing.
In classical electrodynamics, the source of electric field is the electric charge. Source of magnetic field is the electric current. There are no spins in classical electrodynamics. You can have magnetic dipoles, which is what spins ultimately are, but in classical electrodynamics they are represented as tiny current loops.
So why is it an electromagnetic force and not just electric and magnetic? Well, suppose you placed an electron. It's not moving, so it produces only electric field. Now you start moving. Electron is now moving relative to you, so it produces magnetic field. How did it know to produce magnetic field when you started moving? Well, it doesn't. Electric and magnetic fields are aspect of the same electromagnetic field that manifest slightly differently depending on your choice of reference frame.
This understanding is what led to discovery of Special Relativity, which ultimately explained electrodynamics as a relativistic theory. When described in terms of Special Relativity, one usually talks about 4-vector potential as the source of the electromagnetic forces. The source of the 4-vector potential is the 4-current, which includes the information for both static and moving charges for whichever frame is relevant.
This is where we jump off into Quantum Mechanics. In Quantum Mechanics, the charged particles travel as waves, whose behavior is influenced by the 4-potential, giving you electromagnetic forces. The spin is a quantum phenomenon. Particles are point objects, and yet, some of them behave almost as if they were tiny current loops. They have angular momentum and they have a magnetic moment. Working together, they can produce quite significant magnetic forces, such as in the magnet.
Now, so far, in all of the above, the field is just the field. You can have electromagnetic waves, which is light or radio waves, but there is no concept of "photon" in any of that. The field itself is the force carrier. When you apply principles of Quantum Mechanics to the electromagnetic field itself, however, you end up with Quantum Electrodynamics. In it, the field itself is represented as particle exchange. These particles are the photons. The significance of such an interpretation comes from the fact that when you make measurements on the electromagnetic wave, it behaves as if its energy is quantized. You can only absorb or emit a discrete number of these quanta. In other words, only an integer number of photons can be emitted or absorbed.
Of course, superposition makes this distinction moot at any time except measurement. The system can be in a superposition state of having and not having emitted a photon, which is really no different in any way from having emitted half of a photon. And that "half of a photon" can be absorbed by something else, which will receive "half" of the energy, itself going to a superposition. But when you make a measurement, you'll find either a state of whole photon emitted or no photon emitted.
What about force carriers, then? Well, it's not just electromagnetic waves that are quantized, it's the field. The electric and magnetic fields between particles are now described as photon exchange. Now, these photons are virtual, which gives them many odd properties. But they essentially behave the same way. Any interaction can be recorded as superposition of all possible exchanges of integer number of virtual photons. Hence photon is named as the force carrier particle.
There are many mathematical advantages of such a model. Strictly speaking, you do not have to quantize your interaction field. The problem is, it's almost impossible to do any real computations if you don't. And in terms of interpretation, it doesn't matter. The same reality is being described either way. So it ultimately doesn't matter if you want to think of virtual photons carrying interaction force as underlying reality or just a mathematical trick to make our lives easier. There is no real distinction in terms of any observation you could make.
