[Bug]: Installing multiple mods for same server type

### User story

I am a coder and i do server hosting using linuxGSM

### Game

rust and l4d2

### Linux distro

Ubuntu 22.04, Debian 12

### Command

command: mods-install, command: mods-update

### Further information

Came across this issue trying to run two rust servers one using oxide and one on carbon when trying to update the oxide server after just installing carbon it tried to install carbon on that server

Also while setting up a left four dead two server when installing the mods and then running update it installed oxide too that server directory

There is a pull request linked to this that has to fix that adds the server name to the file that gets checked for mods installed #4450

### Relevant log output

_No response_

### Steps to reproduce

1. install a mod on one game instance 1
2. install a mod on one game instance 2
3. Try updating instance 1 mods and get instance 2 mod