Change domain name without changing host (OVH)

One of my clients is re-branding, and we need to change the domain name for their site as their business name has changed. They're staying with the same host (OVH), same exact site (only the logo will be changed). No emails.

I've never done this before and want to make sure I'm doing this right. I've contacted OVH to know what they recommend doing, and it just doesn't sound right?

Here's what they said I should do:

Buy the new domain
Add it via the client's customer account going to 'Hosting > Multisite' following these steps: https://docs.ovh.com/gb/en/hosting/multisites-configuring-multiple-websites/switch targeting the 'www' root folder we're currently using
Go into their Wordpress back-office and change the website's URL there
Regenerate the SSL certificate
It feels like it's missing a few steps?

Edit the database in PhpMyAdmin to reflect the change of domain
Add a 301 redirect via the .htaccess file
Tell Google about the move
Does that sound right? Even then, what happens to the old domain?