Thanks, Mr. Farrington, just the one line on that website gives me a lot more info than I had.
"U.S. vs. ship LADY JOHNSTON 1813 April 17"
So that confirms that it was a capture during the War of 1812, and the adjudication should have details about the capturing ship etc. I'd suspect since the Captain's name is not prominent on the web, at least in my brief search, that he was captain of a privateer rather than a U.S. Navy warship.
Hopefully someone in Philadelphia will be intersted enough in this question to go to the archives you mentioned and retrieve a copy of the proceedings for us. Or, maybe when I get time I'll contact the holding organization and see what they charge for copying that one item. At some places it is very reasonable, like 20 cents a page plus some administrative fee.
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