Difference between Giffen and inferior goods. Why aren't all inferior goods Giffen goods? What is the difference between an inferior good and a Giffen good?
Are the two following definitions for an inferior good equivalent?

Def 1: An inferior good is a good for which the demand decreases after a decrease of its price. 

Def 2: An inferior good is a good for which the income effect leads to a decrease of demand after a relative decrease of its price.

A Giffen good (1) is when after a decrease in price of good (1) the demand for (1) decreases but the demand of some other good (2) increases.

Or is Def 1 just the definition of a Giffen good, which is a special type of inferior good ?

For a Giffen good the Income Effect is strong enough to outweigh the Substitution Effect... But isn't it also the case for all inferior goods? The change in their demand is going to be negative (we consume less after the decrese in price) and that change is equal to the sum of SE and IE, so we also get that Income Effect is strong enough to outweigh the Substitution Effect.
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.