How to determine the direction of induced current flow? There are three ways of inducing current in a loop/coil of wire as shown in my book. We can have a magnet approach a coil of wire, or a wire approaching a magnet. Both can be understood in the same way.

On the other hand, we can also change the magnetic flux by pushing a loop of wire into a magnetic field.

This confuses me. Two vertical lines are cutting across the field. But since they are connected, the induced current, I conjecture, would cancel each other.
I saw in a YouTube video that to determine the direction in such situations as 2, one curl the fingers of their right hand along the wire, with the thumb pointing in the direction of the field. So the curled fingers are in the direction of the current. Basically the directions of current indicated by the thumb is always opposite the direction of the changing field.
 A: The rule is called Lenz's Law.  You already appear to know how to determine the direction of the magnetic field due to a current in a loop, which is part of the answer.  What Lenz's Law tells us is that the direction of the induced current in the loop is such that it "opposes the change in the flux".  

Here's a picture I grabbed from https://web2.ph.utexas.edu/~coker2/index.files/induction.htm to illustrate this.  The B-field is pointing up and increasing.  So we say there is an "induced B field" opposing this increase (so it points down).  The current in the loop is such that it would create this induced field according to the usual right-hand rule for B-fields due to loops.
So imagine that the B-field was decreasing instead.  So if B is up the change in the B-field is down.  Thus the induced B-field would point up and the current would be opposite to what is in the diagram.  Here is another diagram from the same website showing some other cases.  Practice working through the reasoning of Lenz's Law to verify that you see why the induced current is the in the direction indicated for each case:

A word of warning about this concept.  The change in the B-field does not really induce an "induced B-field".  What it really does is create a circular E-field, and it is this which drives the current in the loop.  The "induced B-field" is a fictional construct which makes it easier to figure out the direction of the current.
A: I think the main idea that you are missing is that the induced current is caused by the change in the magnetic "flux" (which is not the same as the magnetic field).  Without resorting to any calculus (which is needed for the real definition of flux) the flux is like the magnetic field times the cross-sectional area that it "goes through" inside your coil/loop.  So there are two key ideas here to be able to understand all cases:


*

*It is the flux, not the field, that matters.

*It is the rate of change of flux, not the absolute size of the flux, that matters.


So, in situation 1 the flux is changing because the field strength inside the loop is changing.  For the magnet approaching the coil the strength of the field inside the coil is increasing, so this makes the flux also decrease.  For the magnet moving away from the coil the strength of the field inside the coil is decreasing, so the flux is also decreasing.  This increasing/decreasing difference is why the induced current direction depends on which way you move the magnet.  Notice that if you hold the magnet stationary no current is induced (remember, it is the change in flux that matters).
Now for situation 2 maybe it is easier to think about having a coil whose diameter you can change.  If you increase the diameter (holding the nearby magnet stationary) then the field strength is the same but the flux increases because the area that the field "goes through" has increased.  So this would have the same effect as increasing the field, such as by moving the magnet closer.  Similarly, decreasing the coil diameter will have the same effect as moving the magnet farther away.
So finally, looking at the scanned picture for your situation 2 we have a coil moving from a place where there is no field to a place where there is a field.  So the flux increases as the loop moves into this region and a current will be induced.  How would this be achieved in practice?  This would be difficult but it would be easy to move a coil from a place with a weak field to a place with a strong field.
Hope that helps!  In my opinion induced currents are hands-down the most difficult things to figure out in elementary electromagnetism, so don't be surprised that you're having some trouble wrapping your head around them.
A: The Maxwell equation says $$\operatorname{curl} \vec E = - {\partial \vec B \over \partial t},$$where $\vec B$ is the magnetic field and $\vec E$ is the induced electric field.That may appear, to an elementary student, to be a bunch of gobbledygook, but it means "the electric field curls clockwise around a change in magnetic field;" normally the + orientation is counterclockwise by the right-hand rule; the negative sign makes it clockwise.
Okay, so we need to first define "magnetic field." One popular way to see the lines of a magnetic field directly to look at the effect of the magnets on iron filings, which will naturally show you some "lines" when you bring a magnet nearby; see these images if you've never seen the effect before. The idea of "magnetic field" is basically just that we're going to take these lines and add an idea of "forwards" or "backwards" to them: these lines therefore "come out" of the North pole of a magnet and "go into" the South pole of a magnet. Please reread this paragraph until that convention is firm in your head. You will know it is when you can appreciate that the Earth's North Pole must be the South pole of some big magnet, which is why the North poles of magnets point North: magnets generally want to align with an existing magnetic field, which means the magnetic field at the surface must point Northward, which means it must be going into Earth's North Pole, which makes it a South magnetic pole.
Now you need to understand the change in magnetic field. The magnetic field has both a strength (how close the field lines are together) and a direction (the direction the iron filings point, combined with the orientation forwards/backwards defined above). If a change increases the strength of the magnetic field, like when you get closer to a bar magnet, we point the change in the same direction as the magnetic field. But if the magnetic field gets weaker, then the change points opposite. If the direction of the magnetic field is changing, then we have to also include a component which points perpendicular to the original magnetic field, pointing in the direction that it changes. The full description of how to do this is known as "vector calculus" and I can only give a couple of basic guidelines on how to calculate these "changes" without that framework.
Now: point your left-hand thumb in the direction of the change of magnetic field: then your fingers curl in the orientation of the induced electric field. It is "clockwise" when you look at your hand thumb-on, or when you look at the change pointing towards you. This means that if a current follows that curling, it goes to a higher voltage; or if it opposes that curling, it goes to a lower voltage.
This same "reverse" rule can also be phrased as Lenz's law. This says that induction works like inertia: changing magnetic fields produce electric fields that would cause a current that would oppose the change. As you may know, a wire also produces a magnetic field. The direction of this magnetic field looks like this: point your right thumb in the direction of the current, then your fingers curl in the direction of the induced magnetic field. If you combine your hands together, pointing your left thumb up to indicate an upward change of magnetic field, then putting your right thumb against your left index finger, you'll see that your right fingers curl into your palm, opposite your left thumb.
Lenz's law gives some awesome quick intuitions. For example: the basic electrical component known as an inductor is a loop of wire usually wrapped around some ferromagnetic material. By Lenz's law, when you try to change the current that's going through it, it induces a voltage which tries to keep the same current going through it. It's like an inertial term, it fights any change in the electron velocity. So as you try to ramp up the current, you fight its voltage; as you try to ramp down the current the same thing happens.  
A: The Fleming right hand rule say that:Hold the thumb,the forefinger and the centre finger of your right-hand at angle to one another.Adjust your hand in such a way that four finger point in the direction of magnetic field,and thumb point in the direction of motion of conductor.
