I don't intend to be mean, but to me it sounds like deep down you have a crisis regarding your 'individuality'. It's like TK said: your appearance is but a reflection of who you are, but it doesn't make or break it if you have to change how you look for a little while. You insist so much on sporting a certain fashion like your life depends on it, that I do wonder if you're trully so confident about yourself as to rely so much on a certain point of your outwards appearance.
Also, you keep searching for answers within the mbti types: "is this type more like this or that type more like that?". In my opinion, the mbti system is something to go by, but not something that 'makes' you; you make it and acknowledge it and not the other way around. However, you seem intent on making it sound like being one type or another would have such a great influence over certain actions, as if it was the type controlling you and not you simply being an individual character.
In all, I do agree that one has to keep their own individuality and persona in a professional setting. Especially in psychotherapy and other psychology fields, because one of the main points about serving people's needs is being 'genuine, natural and yourself'. You can't cause comfort in someone if you're putting on a mask, that much is true.
However, this demonstration of yourself has to come across through your voice, your eyes, your body language... not so much by what you wear or how you comb your hair.
Fair enough, pig tails aren't a big deal. They might provoke a more childish effect, but I don't supose it would cause any distress in the patients. However, if I go into the office and I see that my counsellor has his/her face tattoed, three piercings on each eyebrow, flesh holes on each ear and another one in his lip... well, I'm not saying he can't be the greatest counsellor ever, which he might just be. But the first impression I get is not the one that me -the patient with moral/mental issues, who is probably scared or in conflict or distressed with herself, who took this as a last resource only after much considering- expects to see. It doesn't evoke the sense of familiarity that professionals in this field must have. This type of sittuation can have any effect on the person seeking help: they might withdraw and become unable to totally confide in the counsellor, they might get distracted (as someone mentioned) or they might just get used to it after a while if they can trust the counsellor beyond the appearance.
For each person you'll get a different reaction, but for what it's worth, if you're working to serve people in their darkest hours, and for even the slightest chance that the person might be unsettled by your outwards appearance, you should consider and adapt it. Be yourself, but let others be themselves as well. This is my opinion.