Why don't computers outside see my computer as part of the Cornell network when I'm connected to the VPN?
Not sure I understand your question.
Depending on your VPN, the point of the thing is to give you a relatively secure, private connection to an internal network.
It usually offers a "Tunnel" (probably encrypted) through the level 1 network, and is unidirectional. You may be able to see out into your local network, depending on how the VPN software is configured, but the local network residents should not be able to see anything related to your VPN traffic.
If another computer on the local network CAN see your Cornell traffic, you VPN is seriously defective.
What VPN client do you use? There are clients that block all incoming and outgoing connections to the machine when the VPN tunnel is up.
