Right Hand Rule? I'm trying to determine what the correct right hand rule is, but Google isn't helping. I've seen three different versions of Fleming's Right Hand Rule now, and I'm very confused. 
Here are three different versions I've been seeing:



Would someone be able to explain which is correct and why the wrong ones exist? 
 A: You want the relationship $\vec F=\vec I \times \vec B$ to hold in all cases (direction wise at least. If you were looking at calculating actual numbers you would need to multiply the cross product by a length). You will see that in the first two this is obviously true. In the last one the wording is confusing. It is correct if you take "motion" to be the initial velocity of a positive charge and "current" to be the force, but I'm not sure this word substitution is justified.
Therefore I would not look at the final picture. The first two are valid though.

I prefer the first one based on a fun memory device. Your fingers are like grass in a "field", when you push on something with a force you use your palm, and the velocity or current is the other one (props to someone who can make a good analogy for the thumb being the velocity or current. The best I can think of is hitch hiking?).
A: All the diagrams are correct if they are used in the correct context.  
There are many right-hand rules and the fact that you mention the name Fleming probably contributes to your confusion as is illustrated below.  

You will note that there are two Fleming's rules.  
One for the left hand is for motors and one for the right hand is for generators.
As an aide-memoire the following nomenclature is often used:  
$\Large \rm F$irst finger $\rightarrow$ magnetic $\Large \rm F$ield direction
se$\Large \rm C$ond finger $\rightarrow$ $\Large \rm C$urrent direction
thu$\Large \rm M$b $\rightarrow$ $\Large \rm M$otion implying force for a motor
So your third diagram is the same as my right hand diagram and refers to the rule for generators.  
Your first two diagrams are variations of using the right hand to find the direction of the force for equations like $\mathbf F = \mathbf I \times \mathbf B$ and $\mathbf F =  q\mathbf v \times \mathbf B$ as illustrated below.

A: All three versions are correct. The law that we are looking at here is $f=q v \times B$. Suppose that we have a magnetic field pointing horizontally due north and a proton traveling horizontally due east, then the force is perpendicular to both, so it must be either vertically up or vertically down. So let’s apply all three rules, fixing the field and the velocity and seeing if the rule says that the force will be up or down. 
In the first one the fingers are north, the thumb is east, so the palm is up. 
In the second the middle finger is north, the index finger is east, so the thumb is up. 
In the third one the index finger is north, the thumb is east, so the middle finger is up. (I think that is what this one is saying, but the wording is odd, it may be referring to a completely different law)
All three identify the force as being vertically up, so they all agree and are all correct. 
A: Using your right-hand: point your index finger in the direction of the charge's velocity, v, (recall conventional current). Point your middle finger in the direction of the magnetic field, B. Your thumb now points in the direction of the magnetic force, F(magnetic).
A: The first image says that $(\vec{v}, \vec{B}, \vec{F})$ is a right-oriented orthogonal frame.
The second image says that $(\vec{F}, \vec{v}, \vec{B})$ is a right-oriented orthogonal frame. But $(\vec{F}, \vec{v}, \vec{B})$ is just a cyclic permutation of $(\vec{v}, \vec{B}, \vec{F})$ and orientation is preserved under cyclic permutation. Therefore this is equivalent to the first image.
The third image says that $(\vec{v}, \vec{B}, \vec{I})$ is a right-oriented orthogonal frame. But $\vec{I}$ has the same direction as $\vec{F}$ so this one is also equivalent to the first image.
A: Experimentally, when you have a straight conductor moving in a magnetic field as per above diagram for Fleming's Right Hand Rule, you experience zero current which means that the rule is flawed.
I have set up the experiment and proved it to be so.
