I would like to debate both answers presented so far, but I don't have the rep points to comment yet. If this doesn't belong here, please delete it.
In isaacg's solution, the beginning is not a valid Crossway win by either player since nothing reaches from one end of the board to the other. This was specifically pointed out in the OP "In particular, can you find a win for one player in Crossway that, if you then switched to Mincut rules, would not be a win for the same player in Mincut?" The Crossway game needs to be finished first. Assuming it is O's turn as suggested in the post, they have no valid moves since both open positions create crosscut conditions. Play passes to # and they can play either open position for the win.
00#0 00#0 00#0
# # ## # # ##
0#00 0#00 0#00
###0 ###0 ###0
start win 1 win 2
Play can now continue using Mincut rules. It is now 0's turn again. Regardless of which win condition we examine, 0 has no valid moves. They cannot place a piece or move a piece without creating a crosscut condition. In win condition 1, both of the 0s in R1C2 and R1C4 create crosscut conditions. In win condition 2, the 0 in R3C3 creates a crosscut. Play moves over to # who places the final piece and makes a valid Mincut win.
Moving on to Dr. Xorile's solution. As suggested by isaacg in the comments, red's only valid moves are A6->B5, B4->C3, E4->D3, and F6->E5. But are B4->C3 and E4->D3 really valid moves since they move over the vertical center line? Since there are an even number of squares, I think the closer to the center rule is too vague for this condition. Since Crossway is played on a 19x19 Go board, there is a defined center position that removes the confusion of what moves are allowed. Assuming the B4->C3 move is not legal, red takes the win again via Red A6->B5, Blue C3 (or D3), Red F6->E5#.
If the B4->C3 move is legal, this potentially changes the end result of the game. Red can move B4-C3. Blue places B4 and locks up the win because red cannot place at B5 or move A6->B5 without creating a crosscut. But if Red makes the A6->B5 move, which IMO is the more obvious move to make, the result still ends up with Red as the winner of both games. This is a case where Red wins Crossway but loses Mincut.
Dr. Xorile also postulates swapping the A6 and B4 colors. You can't swap A6 with B4 and E4 with F6 at the same time or Red will have won both games by default. It doesn't matter where blue plays their first turn, they can't play E5 to block the Red win. Blue cannot shift a piece because they can place on either C3 or D3. Blue places either of those, Red shifts F6->E5 and they win again.
It depends on how you interpret the rules.
In the case of Dr. Xorile's puzzle, it boils down to how you interpret the closer to the center rule on a board with an even number of squares. In all of the other tests I made, I could not find a condition in which either side won without having the crosscut advantage. That is to say that once a Crossway line is made, the opponent cannot play to block both a vertical and horizontal line without creating a crosscut of their own. If the Crossway winner wasn't immediately able to place the pertinent piece to win, they only had to bide their time until the opponent shifted a piece out of the way allowing the win. The crosscut rule always seems to favor the Crossway winner. My testing assumed that a piece could not move over a vertical or horizontal center line to be considered closer to the center.