How do I run 2 servers from the same install, one modded and one vanilla?
On the modded I have set identity as "battlefield" and +oxide.directory "server/battlefield/oxide".
But when I run a vanilla from the same installation Oxide and Rust:IO are running.

Extensions will always load, they aren't plugins and aren't instanced.

You don't mention which OS your server is on, but on my Linux box I simply run them on two separate user accounts and give them different port numbers, job done.

Windows. How do you make the vanilla server not load Oxide and Rust:IO?

Windows - no idea, sorry :(

runescape

Windows. How do you make the vanilla server not load Oxide and Rust:IO?

You're sharing an install, there's no option not to load them as it's already modified.