The first explanation is correct.
The charged particle (or object) does not experience a resultant force due to its own magnetic field, it only experiences a resultant force due to the applied magnetic field. Any forces the particle (object) exerts on itself are internal forces, which do not affect the motion of its centre of mass. This applies also to electrostatic and gravitational forces, and is a consequence of Newton's 3rd Law.
However, the magnetic force on another particle or object is the sum of the forces due to the applied field and the field generated by your moving charge.
Likewise both charged plates of a capacitor contribute to the total electrostatic field between them, but the force on one plate is due only to the field created by the other charged plate, not due to the total field between the plates, which includes its own electric field. See Attractive force between capacitor plates. By contrast, a charged particle between the plates experiences a resultant force due to the electric fields of both plates - but again not due to its own electric field.
Note that I am assuming that there is no acceleration of the charged particle. As is explained in Does a point charge exert force on itself?, internal forces do not cancel out when there is acceleration, leading to the emission of energy.