The changing flux in the core creates a 'volts per turn' around the core. This is the voltage that perfectly opposes the input voltage on the primary, and generates the secondary output voltage.
When a secondary current flows in phase with this voltage, power is abstracted. The same power is delivered by the primary current flowing in phase (or to keep sign convention with the secondary current flow, anti-phase) with this voltage.
So my question is: What exactly changes when we connect a load to the transformer, which results in power transfer from primary to secondary?
The primary and secondary currents change. We know that a current flow through a voltage difference moves energy.
In short, which magnetic parameter is responsible for power transfer between primary and secondary?
The only thing connecting the primary and the secondary is the magnetic, or electromagnetic, field coupling both windings. It is therefore the field.
At this point, you need to examine what you are actually looking for. Do you want an equation which predicts the magnitude of the various effects, or do you want to know why they are happening.
Most engineers would be happy with equations to predict how ideal, and non-ideal, transformers behave. In that sense, your question might be better asked on the physics stack, where they are interested in more fundamental things.
But wait, physicists only produce equations to predict what's going to happen, an elaborate method of book-keeping. For instance power flow in electromagnetism is described in terms of the Poynting Vector, the cross product of voltage and current, which actually identifies that power flow occurs not inside copper wire, but in the fields in space surrounding them. So if you dip into physics, you find that power flow in a wire is already non-intuitive, and you haven't got to anything as complicated as a transformer yet.
Maxwell's equations will tell you what changes in current and field will produce other changes in field and current, but not why. Quantum mechanics will tell you what's likely to happen when you set up an experiment involving charged particles but not why, absolutely certainly not why. And then those particles are best thought of as excitations in fields anyway.
At the bottom, there's 'If it Happens, it Must be True'. Ideal transformer equations, Maxwell's equations, the Poynting Vector, all described what we see, experimentally. Live with the transformer equations long enough so that your intuition aligns with what happens, and then you'll intuitively get them.
Consider a mechanical analogy. Take two magnets and align them same poles facing, so they repel. Push the first magnet, and let the second magnet move. When you push the first magnet, you do work on it, it does work on the field, the field does work on the second magnet, which moves away and does work on something else. You haven't stored any energy in the field, and the energy has somehow been transferred from the first to the second magnet, through the field. The field could easily be constant in strength and the magnet spacing stay the same during this operation.
