No. The original Xbox System Link cable is a standard CAT5/5e Ethernet crossover cable (or regular patch cable with a crossover adapter). The Xbox 360 uses standard straight-through Ethernet cables (or direct LAN/Wi-Fi) for System Link.
You cannot plug the original Xbox cable into the 360's Ethernet port and expect System Link to work between an original Xbox and a 360—cross-generational System Link is not supported (different network protocols, XDK versions, title IDs).
For 360-to-360 System Link: Use a standard Ethernet cable (or router/switch) and enable System Link in-game. Both consoles must run the exact same game version / title update.