Question: One of the main problems I had to deal with, when I started to meditate and study the Dhamma, was the consequence of giving up attachment to reputation and pride.

Of course this is an extremelly positive thing to do, you simply become stronger when you stop caring about how other people see you, you just care about your actions, about karma, about your thoughts.

The problem is that people think you just don't care about anything else, they think you are not commited to your job anymore, to your appearance, to other people feelings... for instance: If they give you a bad feedback and you don't suffer and don't react to it, they think you are ignoring them.

It looks like we need small portions of drama to live in this society, non attachment to reputation makes you look like an alien or a brain washed person. I don't know how to balance that sometimes, any advise?

Answer: 
In buddhism, what is referred as attachment is the type of attachment which keeps wheel of dependent origination.

In each case when a through arrives in your mind you get a sensation. When you react to this sensation with liking then it is attachement. If you do it with dislike it is aversion. All this because you do not understand the true nature of things.

If you experience any sensation with respect to thoughts of responsibility to which you react then there is attachment to reputation. If you do not react to the self awareness and reputation then there is no attachment. You can have applied through without attachment. This is more emotionally balanced than you doing things for stimulus from thoughts on reputation. This is more like a carrot and stick approach at a subtler level.
The right fix, in my personal opinion, would be to respond as best as you can when people tell you such things. Try to see what it is about your reaction that gets on these people's nerves. It's not the case that you have to suffer when making use of social norms and conventions in order to maintain a dignified conversation. Try to see if you can find a way to appear right and normal to these people, without sacrificing your stability.

May I also suggest that, so long as you're not a monk, you don't want to sacrifice your job and relationships for the sake of not wanting to engage in things you don't like just right now. It's the long term well being that you're better off working on - and getting fired or thought lowly of by your friends & relatives isn't going to be part of that. Surely you can find a way to balance things out rightly.

Best of luck.