The two situations where time zones are relevant is a very good observation, but I still think we should optimize for the more common case, which is communicating across timezones.
So, it's better to keep the second problem ('what time is morning?') more accessible, since you need that in both cases.
The current use of timezones seems primarily designed to ease adjustment for those traveling between them. Work starts at 9 AM in region-X, you move to region-Y, adjust your time and work starts at the "same" time.
Today I think however there are a lot more cases of cross-timezone communications, and a lot of people are telecommuting to work and various events. It may be time to move on and adjust to better support distance communication and telecommuting.
The national networks can advertise that a popular show comes on at 8 pm, and it will everywhere.