I discovered a couple of problems, which actually caused the time (for me) to be displayed correctly (Eastern Time Zone), while the GMT was being displayed as -5.
The first problem was that vBulletin was configured with Eastern Time Zone when it should have been Central. The server is located in Dallas, Texas, which is located in the Central Time Zone.
A second vBulletin option allows daylight savings to be enabled or disabled. It was set to yes (enabled), which is correct for Dallas.
The current time zone offset in Dallas is GMT -5, which will change to GMT -6 when daylight savings time ends on October 30.
The second problem was that in my control panel settings (Edit Options) the DST Correction Option (under the Date & Time options) was set to "DST corrections always off." I changed that setting to "automatically detect DST corrections."
After the corrections, the time at the bottom of the screen after this message was posted reads:
All times are GMT -4. The time now is 07:24 AM.