is losing a shield considered taking damage ?

Taking damage yes (for something like Hate), being damaged no (for something like Gideon Hask).

Depends how you lose it.

If you lose it to Dalan Oberos (Kimogila)'s ability, no, if you lose it due to Inertial Dampeners, no, if you spend it to recharge your Bomblet Generator, no. Otherwise, yes.

And yes, as noted, the Damaged keyword means having at last one damage card assigned to your ship, and has no bearing on whether you have Suffered Damage or not.

loosing a shield is not taking damage, no. the effects are not the same.

loosing a shield can happen in several different ways, such as spending it for inertial dampeners, bomblet generator or miranda donis pilot ability.

you can also resolve the effect of taking damage by loosing a shield.