What is the difference between caramelized onions and "crispy onions"?
Both crispy and caramelized onions are cooked for a long time, and will be very brown.  However, they are cooked slightly differently.

Caramelized onions are usually cross-cut on the onion to release its moisture, and then cooked over very low heat in a crowded pan, stirring infrequently, so that they gradually release their sugars and liquid and it turns to caramel.  Depending on the onions and desired result, you may even cover them, an add a little liquid and/or sugar.  The end result is very soft and very sweet.

Arabic-style crispy onions are cut pole-to-pole in order to avoid rupturing cells in the onions.  They are then fried over medium heat in an uncrowded pan, stirring regularly.  This lets them dry out and become brown and crispy, even burning on thin ends.  These onions should be a mix of crispy and chewy, and more savory than sweet.