Here's a quick lesson on arcs in animation. Everything moves in an arc. If it doesn't move in an arc it's wrong (again, unless you know what you're doing). The human body moves very much in arcs, and it's probably not more prevalent than in the arms. This is a very, very, very crude representation of something in that book I told you about:

On the top-left is a pendulum (think grandfather clock swinging thingy). When it moves the weight (circle part) moves at an arc, because where it pivots is stuck in one spot. Right now your arm movements are doing what the bottom left drawing is which is the wrong thing to do. When the weight moves it just goes straight across. If you think about it for a minute you'll understand why this is wrong.
...Pencils down. Let's see your answer!
If you said that it's because when the center pendulum is in the center position, the suspension (the long stick holding the weight) changes size, then you are correct! Obviously, a static solid object cannot change it's size, thus the pendulum
has to move in an arc.
The super excellent right hand drawing is supposed to be an arm (if you squint hard enough

), which oddly enough looks like a pendulum! The top circle is the shoulder, or the pivot. The second circle down is the elbow, or the weight of the pendulum. Now look at your arm. Go ahead, don't be shy! Now move it every which way you can think of. Does it ever, even once, change it's length from shoulder to elbow? I will just assume you are a normal person and that your answer is no, it in fact does not change in size. So from this little experiment, which of the two left hand drawings will more closely imitate the actual swinging of your arm? Yes, you are correct.
To further illustrate the point, I've taken the liberty of drawing the forearm down to the hand in the pendulum/arm exercise. You'll notice that the wrist is also moving in an arc, and the hands move in an arc. Now think about why this is. If you need a hint just read the above paragraph.

So, in answer to why your characters' arms look funny, it's because they are sliding along the same plane, rather than swinging from a pivot. Does this help?
If you're wondering why I didn't address this in my own edit of your animation its because I forgot about it.
