Quiet extroverted or loudly introverted is possible in an enviroment where people have different concepts of what E-I means. In a very basic matter, there are at least 3 definitions of extroversion/introversion over MBTI unofficial internet community, and in short they are these:
1) How sociable and outgoing the people is.
2) If gather "energy" from the "inside" or the "outside".
3) Preferency towards the object - if attention is towards the object (external) then its extroversion, if attention is towards self (inside) then its introversion. Cognitive functions use this definition.
Just a quick example, if you are watching the sunset you are doing an introvert activity in 1 and extroverted activity in 2 and 3. So, its possible to be introverted in one and extroverted in another definition, making the person being a quiet extroverted or sociable introvert.
Also, the E-I spectrum isnt "you are I" or "you are E", they are not extreme, its possible to be between them. Even when there is a board marking "E are this, this this and that" with "I are this, this this and that" side to side, its possible to be ambiverted as well.
Also, the guy that answers this thread more quickly gave a link to a good read on that.
And for last, you are not alone, search ambiversion MBTI on google and you might find some stuff (although some of them will be defending that ENFP and ENTP are the ambiverted types). Ambivalence is possible in any of E-I,S-N,F-T or P-J dimensions.