Do points in a wave move along with the wave? I know in a standing wave, the particles just oscillate up and down. 
And in longitudinal waves, particles vibrate back and forth parallel to the direction in which wave is moving.
But I'm confused, do the particles/ points in a wave travel with a wave in a travelling wave? 

And also what is meant by "In traveling waves, the amplitude is the same for all particles along the wave"?
 A: A wave is often a disturbance that travels through a medium, leaving the medium as it was. For example, a wave on a string moves each part of the string sideways. After the wave has passed, the string hasn't changed. 
Sound is a pressure wave. Air vibrates back and forth in the direction the wave moves. 
In water waves, each part of the water moves in a circle. 
Wave motion is not the only kind of motion. For example, wind and water currents do transport air and water. But we don't call that kind of motion a wave. 
A surfer can ride a wave. But he isn't part of the wave. He isn't undergoing wave like motion. 

Some waves don't have a medium. E.G. light can travel through vacuum. Even though it is similar in some ways to a wave on a string, it is not the same thing. 
A light wave describes the force a vibrating charge can exert on anther charge. Because the source charge moves back and forth, the electric forces push the other charge back and forth. Nothing is waving in between the charges. 
The equation that tells you what force to expect at a given place and time is much the same as equation that tells you how far a piece of string is from its rest position at a given place and time. 
A: No, they don't.
At least if by "points" you mean material particles. A specific location of the shape of the wave (another possible interpretation of "point in a wave") moves with the wave by definition.
A traveling wave is a wave which propagates, i.e., a non-standing wave. In a standing wave there are points which move the most (highest amplitude), points which don't move at all (zero amplitude), and points with amplitudes in between. For a traveling wave that doesn't disperse the amplitude of movement of the affected particles is the same for all of them. That's what is meant by your quote.
In the original post picture, that's illustrated in a transversal wave, like the waves in a rope - note the small circles demonstrating the different amplitudes in a standing wave and the constant amplitude in the propagating one.
(I'm restricting the answer to material waves.)
A: In simple terms you can think of a mechanical wave as something which can transport energy and momentum without the medium through which the wave is travelling moving bodily with the wave, rather the medium just oscillates about a mean position.  
So you last animated graph is showing some of the particles which make up the medium oscillating about a mean position and in phase, so they are one wavelength apart (50 m).  
The period of the particles is 20 seconds and so their frequency is 0.05 hertz so the speed of the wave is 50 $\times$ 0.05 = 2.5 m/s.
This is the speed that a crest or a trough moves which we call the speed of the wave.
You will note in that animation that the maximum excursion from the equilibrium position of each particle is $\pm $ 6 m and so the amplitude of the wave is 6 m.  
You might look at the PhET simulation Waves on a string with "no end" and "no damping" which illustrates these ideas.  
Then if you introduce a "fixed end" or a "loose end" and still have "no damping" you will see how two travelling waves can form a standing wave.
Note that you will have to adjust the frequency of the oscillator to get a standing wave pattern.
