Note that a current carrying wire produces a circular magnetic field that's why it doesn't matter how you hold your hand ie how you rotate your hand around your arm as long as your thumb shows the direction of the current.
Edit after comments:
See the illustration I've added below. Now use your hand in the way that you've learned and convince yourself that what I've drawn below is correct.
![enter image description here](https://i.stack.imgur.com/buf8K.png)
When the loop is wholly within the region of the uniform magnetic field, then there is no induced current even when the loop is moving. As you pointed out, in this case the induced currents on opposite sides of the loop are opposite and equal so they cancel out.
It is only when one side of the loop leaves the region of the magnetic field, or more generally enters a region in which the field is different, that the induced currents are no longer equal. In this case there is no induced current in the part of the loop which is no longer within the magnetic field. More generally, the side of the loop on which the magnetic field is stronger will determine the direction of the resultant current.
If the loop is moving out of the region of magnetic field with a uniform speed, would the induced current be steady? If not, why? Because of the loop's geometry?
Good follow-up question.
Induced emf (and therefore also induced current) is equal to rate of change of flux linkage through the loop. If there is a sudden change in flux linkage, the current will change suddenly.
![enter image description here](https://i.stack.imgur.com/ITUaI.png)
If the loop is a rectangle with one side parallel to the edge of the magnetic field, then there is a sudden change in flux linkage as that edge crosses the boundary, and again when the trailing edge crosses. In between there is a constant decrease in flux linkage, therefore a constant current. The current-time graph is rectangular.
For both a circular loop and a rectangular loop with a corner crossing the boundary first, the current will increase and decrease continuously from zero to a maximum, because the flux linkage is not changing suddenly. For the circle the induced current does not change uniformly, because of the non-linear edges; the current-time graph is semi-eliptical. For the rectangular corner=first loop the current-time graph is triangular or trapezoidal. For both circular and rectangular corner-first loops, the the maximum current occurs when the widest part of the loop crosses the boundary, because that is when the flux changes most rapidly.
Best Answer
Can you clarify what it is about the diagram you find confusing? I think maybe you are confused because the picture shows (via x's and dots) the magnetic field in the plane of the loop. Perhaps you are confused about "inside and outside" the loop because you are trying to picture this scenario in three dimension. Try imagining this loop is floating on some water. If you try using the right hand rule is there a region where you will always be dipping your fingers into the water? and a complementary region where your fingers will always be coming out of the water? Don't worry too much about what happens well above and well beneath the surface of the water, although that is also something you should try to understand eventually.