Yes, your understanding is almost correct!
No need to install it on each Hyper-V server, if you do it, it would be on-host proxy servers, not off-host.
To use off-host backup proxy, you need to configure it in the same way you did it with the production host, meaning that these steps should be done:
1. Install off-host backup proxy on a Windows Server 2008R2 (given that all your production hosts are running Windows Server 2008R2)
2. Enable Hyper-V role on this server
3. Deploy hardware VSS provider on both production and off-host proxy server
4. Present your SAN storage in the same way you did it with production Hyper-V hosts.