You have pretty well understood it. Be sure your two sites are NOT on the same machine and have different IP addresses.
the URL is the SAME for both sights -the difference is in the DNS (nameserver) which is controlled within your domain nameservers info at your registrar.
And you have picked up on the syncing issue. As to what "big companies" do, there are several solutions. from simple to complex.
What kind of volume to you have? That will help you decide how much effort and cost you need to insure your info is relatively in sync.
Various solutions - you can move your DB to one different location and let both app locations write to the same DB - then make more frequent backups of the DB in a location other than your site location. eg. if you have Softaculous you can set your backup for just the DB and save the backup on a different location like a dropbox account or AWS if you use that.
Look into what your backup options are from your hosting company - are you having a daily backup with several generations auto ran and stored off your site server??
Your hosting company should able to provide you more specific options that work within their framework
Do you get notification when your site went offline?
There is also the ability to have a hot-swap raid backup on some hosts - that does cost more, but what it is two drives assigned to your account - and they both get written to almost simultaneously, if the main one goes down, the hot swap is actually pulled physically and placed in a new location - most host who provide this monitor your site and if there is a mechanically failure of a drive they do the swap quickly.
