You can customize how the subscriber and team member names are displayed in communications, as well as how automated trigger messages display in the conversation threads. Unified Inbox display settings can be controlled from Inbox > Inbox Settings...
Unified Inbox Settings
Choose to use either the company name or subscriber name in guest communications. Only one of these options can be toggled on.
Shorten team member last name to first initial only: Depersonalize the communications by showing only the first initial of the team members last name when communicating with guests.
Display trigger messages collapsed by default: Some trigger messages can get quite long and make it difficult to follow the message history for a conversation. This setting allows you to collapse trigger messages when shown in the Inbox, making the conversations easier to follow.
Trigger creates last message received event: This option (if set to OFF) means that trigger messages will not update the last message timestamps, move the conversation to the top of the list, or change the last message preview text.