That's a very interesting question!
The logo of AlJazeera is a calligraphic interpretation of the word 'Aljazeera' in Arabic. The logo can be read by native eyes fairly easily. But this is only a sign of a 'good' and functional design. The same word could be rendered completely unreadable by another designer. Case in point, your second picture. That picture is, of-course, a horse. It is formed by Arabic letter forms. I'm pretty sure it is also contain coherent words. But if the intention was to be readable, it has failed the intention. Otherwise it is generally a common practice to use alphabetic forms to produce a visual form which is not meant to be read. I should add I do not understand Arabic and for a native Arab it may be somewhat easier to decipher the words hidden in the horse. But generally when you go that far, you really are not writing something to be read! It's meant to be seen.
Oh by the way, the first lines disappearing in the first animation are mostly ornamental but not randomly placed. Those lines are the extension of the extremity of several letters. They are used to create a visual balance in the calligraphy. But since they have a 'legitimate' presence, they do not diminish the ability of a native eye to read the word in the logo.

Note that the wikipedia animation is misleading, the logo is based on http://Diwani calligraphy and thus the shapes are pretty normal relative to that specific style of Arabic calligraphy which is quite different from the simplified Arabic at the binging of that animation (seriously, who in his right mind would compare a Diwani based logo to Arial!).

If you want to design an arabic calligraphic logo I think you need somebody who is able to read arabic and one calligrapher. there are no rules, It's just about understanding of rhythm, design and deep knowledge of the writing system, either it's Nasta'liq or Diwani. second picture is an ornamental calligraphy (similar to tughra) and not a subtle one in my opinion and I agree with khaled, this animation is very bad way of interpreting the logo.

some good examples in Persian/arabic:
http://www.qoqnoos.com/body/graphic/JABARI/p02.jpg
http://www.honar.ac.ir/Farhangestan/images/Arm_Farhangestan.JPG
http://nowrozkhani.persiangig.com/image/Jabbari/Jabbari%202.jpg
http://www.qoqnoos.com/body/graphic/JABARI/p01.jpg
http://www.qoqnoos.com/body/graphic/JABARI/p08.gif
http://www.asriran.com/files/fa/news/1388/11/5/125504_965.jpg

Bahman.E

Too bad you can't read Arabic. I have written an article with some analysis and useful links on the Tugra, which is a calligraphic device that is the basis of this logo. The article is in Arabic, but do check it out. You can also google the word 'tugra'. You will find a lot of material that will help.

Khalid

PS: It's nice to know that we have Behnam and Bahman here. I used to work with an official in the Iraqi government called Behnam. International Consultants called him Benham. Probably, yet another good example of 'unity and variety'. And, Alch, please update your profile and tell us more about who you are. Finally, the calligraphic style in the Al-Jazeera logo is Diwani.