Is there an option to change the user-facing ticket statuses?

The default ticket status values available in user-facing are:

Open - A ticket that has been opened by an agent or client. The first response time is calculated from the timestamp when the ticket is created.

Closed - A ticket that is closed and on which no work is being done; normally after being acknowledged by the client for closure. The resolution time is calculated from the timestamp when the ticket is created until the timestamp when the ticket is closed.

Pending - The ticket is awaiting a response from another vendor, admin or developer, not the client or agent. The on-hold time is added to the client's wait time and not agent wait time. On-hold is set where there is no predictable estimate time for a reply.

In Progress - A ticket on which work is currently being done.

Resolved - A ticket on which work has been finalised and marked as solved by the agent.

Reopened - A ticket which has been reopened.
The above standard ticket statuses cannot be edited; however, you may wish to add additional statuses based on your ticket lifecycle or workflow.

Awaiting Customer Response - A ticket is waiting for a reply or action from the client.

Awaiting 3rd Party Response - A ticket is waiting for a reply from a vendor, developer, or service provider for approval. The agent contacts the 3rd party and gets the approval for the client's request, then updates the ticket.

The above two statuses are editable and can be enabled or disabled based on your requirements. For these two statuses, the user-facing label can be changed to a custom phrase.