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: 
I too have felt some resistance, although it did eventually resolve. For one, try being more compassionate. For a long while I simply desired wisdom, and did so in a way that was neither mindful of the consequences of my actions(and how people felt about them) and had a sense of pride. Now I am practising more compassion, and its harder for negativity to phase me. Consequently, I am not trying to shield my ego from the negativity, instead I embrace it. Through companions sharing this, they provide insight into how I could help them better. It was only through realising how my actions were harmful(atleast from the perspective of my friends) to the relationships, that I was able to eliminate the misunderstanding. 

If you confront the ignorance, rather than purely ignore it, and practise compassion, you may find a more harmoneuous relationship. Of course there are people who will resist changing their attitude, such as someone who may be biased against Buddhism or any religious practise! In this case, try and see it from their perspective, and act in such a way to minimise their suffering. If they have no reason to suffer, your suffering should too reduce. 

I used to try and impose my attitude towards vulgar language upon others, but only through accepting the present nature of some people, and their use of such language, was I able to eliminate my suffering. I let go of my attitude, and it no longer bothers me. 

Good luck, hope you manage to find the trickly balance between practising yourself and being understood by others.
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.