Older textbooks often attributed the force on a current-carrying conductor in an external magnetic field to an interaction between the external field and the conductor's field.
The idea goes back to Faraday, who conceived of magnetic field lines as 'lines of force', under tension and pushing out sideways. Think of the pattern of field lines between two bar magnets placed in line, with the North Pole of one facing the South Pole of the other with a gap between them. Can't you just feel those lines of force pulling the magnets together? Even more interesting is the resultant field pattern that you get for the set-up you've shown in your diagram. Many of the resultant field lines (co-inciding at each point with the direction of the resultant field of wire and magnet) run in curved paths from the N to the S of the magnet. But the lines are closer together (as the fields re-inforce) above the wire than they are underneath it (where the fields tend to cancel). So the lines above, under tension and pushing out sideways, act like catapult strings, forcing the wire downwards, in accordance with your Fleming's left hand rule!
Maxwell developed the theory of Faraday's lines of force mathematically. The Maxwell stress tensor, defined for each point in a magnetic &/or electric field, was the outcome. I believe that the motor effect force on the wire can be correctly predicted using this concept.
For simple set-ups such as yours, it is, of course, far easier to consider the force to be due to a direct interaction between the magnetic field and moving charges in the wire – the good old Lorentz force law.