I'm not sure where this came from!
In the original case, you provided the force to the right, which moved the wire to the right through the magnetic field into the page. That created an emf upwards, which could cause a conventional current upwards.
Now that upward conventional current in the into-the-page magnetic field causes a force on the wire directed to the left.
Only one force is created by the current - to the left.
As to the magnitude of the force, that depends on the field and the motion, so is not obviously related to the original force you applied to the wire.
If you took a stationary wire and applied a constant force, the wire would accelerate, producing an increasing emf and (depending on the circuit the wire was in) an increasing current. So the force would be increasing in opposition to the motion, reducing the acceleration, until there was equilibrium with the wire moving at a constant speed with the electromagnetic force exactly balancing the constant applied force.
(In other circumstances, with a varying applied force, the electromagnetic force could exceed the applied force and even be in the same direction as the applied force at some times.)
I was puzzled by your comment that you had been taught that the LHR used conventional current and the RHR used electron flow. This seemed incredible to me. Although I've never used either rule (I would get mixed up which was which), I remember being taught that thuMb was motion, First finger field and seCond finger current and to me current has always meant conventional current. A conventional current may be caused by flow of electrons, holes, ions or whatever, but always is from positive to negative.
In fairness to you I thought I'd search the web to see what other people said and was utterly amazed to find a website which said,
http://www.education.com/science-fair/article/effect-magnet-electron-beam-right/ (there may be others, but having found one, I stopped looking) so it is understandable that people get confused! All I can say is that, after more than half a century of doing electronics, I've never used or thought of current as anything other than the conventional flow from positive to negative and never come across anyone seriously using any other convention. Generally it is only mentioned to explain that it is an arbitrary convention (or in weird discussions like this!) If I need to talk about electrons, holes or ions, I am careful to say electron flow, ion flow or hole movement or something like that and reserve the word current for the associated conventional current.As a general point, there are other situations in science where such arbitrary conventions are used, some much less widely accepted than this one. It never matters which one you choose to follow (except that you may confuse other people), but you should always stick to the one you've chosen. Changing conventions for different parts of a calculation or argument, is a recipe for errors. If you ever choose to use a convention which is not as widely accepted as this one, it would be a good idea to state your chosen convention clearly at the start.
