Difference between Giffen and inferior goods. Why aren't all inferior goods Giffen goods?
Def 1 is wrong. An inferior good is a good for which the demand decreases after a decrease in the agent's income. (not a decrease in the price of the good). 

Definition 2 is trying to define the same concept, "an inferior good" so it is also wrong. Instead, both definitions would be appropriate if they were describing a Giffen good. 

Giffen goods are indeed a special case of an inferior good. These are goods for which the law of demand does not apply. Since their demand has a positive relationship with price. The reason why this positive relationship is possible is that when prices change, they create income and substitution effect, and when goods are inferior, these effects go in the opposite direction. Thus, if the income effect dominates the substitution effect, you get the (seemingly counter-intuitive) behavior that demand increases when prices go up, and decreases when prices go down (a Giffen good). Note that to be a Giffen good you need to be inferior, but the reverse is not necessarily true, for example, if the substitution effect dominates the income effect.