Difference in Lorentz force?
Let start by defining the variables in your equation: \$ F=IL\times B \$

-I is the current, within your field

-L is the Length of your conductor in the direction of current flow

-B is the magnetic field (magnetic flux density, more correctly)

Due to the cross product and the right hand rule, we know that the direction of force is to the right. 

You're intuition is correct, the force in the second diagram is less, but not because L is smaller, but rather because not all your current is within the influence of the field. 

If we assume that the current is uniformly distributed throughout the width of the wire then the force on the wire will be proportional to the the width of the wire within the influence of the field. 

That is, if 20% of the wire's width is within the field, then the force on the wire will be 20% as much as it would have been had the whole wire been in the field. 

\$ F=\frac {w'}{w}F_0 \$ 

where \$F_0\$ is the force in the first diagram and \$ \frac {w'}{w} \$ is the ratio of the width of the wire within the field.