This is the original puzzle. Notice that some of the tallies are missing -- the puzzle can still be solved without them.
First, let's fill in all obvious water cells. Any row or column with a tally of zero can be filled in. Likewise, the right-end segment is mostly surrounded by waters.
The right-side ship end obviously has a segment to its left. Inserting a wildcard segment, and adding the obvious waters, we get the next frame.
The top row has a tally of five, and there are exactly five non-water cells remaining. Therefore, the entire row consists of ship segments.
Now here's where Wraparound Battleships gets interesting: we need to take the board's torus property into consideration.
First, that's a battleship on the first row! The three segments on the left wrap around and connect to the single segment on the right-hand side.
Second, which cells around the newly inserted ship segments can be filled in with water? Because the board is a torus, the battleship at the top will cause cells at the bottom of the board to be filled with water! The next frame will show the cells that can be filled with water. Those deriving from the torus shape are circled in red.
Good! The bottom row can be filled in with ship segments. Filling in all obvious water cells gives us the next frame.
The next-to-last row can be filled with ship segments. Since we have already located the battleships, the fifth column from the left contains a wrapped-around cruiser. Note also the wrapped-around destroyer in the next-to-bottom row.
The third row from the bottom can be filled with ship segments.
Since we have found all three destroyers, the circled ship cannot be a destroyer. Having already found the battleship means the circled segments are a cruiser.
The third column from the left can be filled with ship segments.
This allows the fifth row from the top to be filled with water.
The last remain ship, a submarine, can be filled in the third column from the right. All remaining cells are water.
Q.E.D.