Sysadmin and DevOps
Responsible for the site being reachable, fast, and recoverable after failure. Their work is invisible by definition: you notice it only when it is missing.
When your project needs this person
For any site that generates revenue. On a small site the hosting provider covers the role, as long as you know exactly who and what it covers.
What happens when they are missing
There are no backups, or there are but nobody tested them. The outage happens on the busiest day and recovery takes a week instead of an hour. The second most common consequence is a slow site with nobody able to say why.
How this appears in an offer
Appears as "hosting", "maintenance" or "SLA". Ask one thing: how far back do backups go and when was a restore last tested.
What they do NOT do
Does not build the site and is not responsible for what it says.