An example backup script for Minecraft servers on Linux. Requires the zip utility.

backup.sh

#!/bin/bash

# Set the script to stop when an error occurs.
set -eo pipefail

# Get the current date as a variable
printf -v DATE '%(%m-%d-%Y)T'

# Get the server's directory location and make sure we're in that directory
SERVER_DIR_PATH=$(dirname "$(realpath $0)")
cd $SERVER_DIR_PATH

# Script variables. Customize to your liking.
BACKUP_FILE_NAME="Server $DATE Backup"
## $PWD will put the backups folder in the server directory. If you want
## to provide a directory outside of the server's directory, put the
## absolute path to the directory here.
BACKUP_DIRECTORY="$PWD/backups/"

# Announce server backup in console
echo "Started server backup..."

# Append the file extension to the file name and create the counter var
NAME="${BACKUP_FILE_NAME}.zip"
COUNTER=1

# Create the backups directory if it doesn't exist
echo "Checking for backup directory..."
if [ ! -d "$BACKUP_DIRECTORY" ]; then
  echo "Backups directory does not exist. Creating folder..."
  mkdir -p "$BACKUP_DIRECTORY"
fi

# Now, check if a file with the current name already exists inside the backups directory
# If it does, a while loop will increment the counter and create a new filename until a unique one is found
while [ -f "$BACKUP_DIRECTORY/$NAME" ]; do
  NAME="${BACKUP_FILE_NAME}-${COUNTER}.zip"
  COUNTER=$((COUNTER + 1))
done

# Get the name of this script
SCRIPT_NAME=$(basename "$0")

# Compress the files, ignoring this script and the backup directory
zip -r -q "$NAME" * -x "$(basename "$BACKUP_DIRECTORY/")/*" "$SCRIPT_NAME"

# And move the archive to the backups directory
mv "$NAME" "$BACKUP_DIRECTORY"

echo "Backup completed! File located at \"$BACKUP_DIRECTORY/$NAME\"."

Updated. The script will now archive the files in a directory layout relative to the server's directory and not the root directory, it will create a new backup every time the script is used, and it will ignore the backups directory in the archive.

Ha, that was all what I was also modifying it to fix, got mostly the same results, but the counter is a good idea that I hadn't thought of.
Thanks for the update! The error check at the top is nice too :D