No. While the DB9 has nine pins, typically only six of them are wired. The pins usually not used are not connected. From the pinouts, you can see the difference is rather big, and you would need a joystick cable with all nine pins wired.
What you could do however is cut off :-( the moulded connector, and solder on a new DB9 connector, carefully according to the other pinout. Bear in mind though that the TI pinout refers to two outgoing wires for testing the joystick (that it is connected?) and we'd need to figure out when and how these wires are used. Perhaps a dedicated TI forum/community could help out. Once we know how a TI joystick is supposed to handle and respond to these signals, we can move on. The Atari compatible Wico joystick is a "dumb" device which simply short circuits signals as you wiggle the stick or press the button. I'm sure the joystick functionality on the TI is technically the same, but that there might be more to it than that.