Personally, I would say, "Ow/Sh*t/Cr*p, my hand!" It is not necessary to exclaim the nature of the injury, unless one is to describe it to somebody else, in which case one would be more likely to say, "Ah! I've just trapped my hand in the door!"
'Battered' is the wrong term to use here, because one would not usually say this of getting one's hand stuck in a door; it would be more appropriate in the context of continous blows.
I hope this has helped you to understand a little better.
I think that "crushed", "smashed", "hurt" etc. are better choices. "Battered usually has the connotation of being beaten up."
I always have a problem with "battered". I go into a fish and chips fast food place and they sell "battered shrimp". And I think, "Who is the sicko who goes around beating the hell out of poor, innocent shrimp"?
I think that everyone really does make the distinction between "battered" (deep fried) and "battered" (beaten up), but I think better words are available and you should pick one that more nearly expresses your thoughts. In this case I think "crushed" is the better word.