(We've crossed swords on the subject before)
1)know what the project is
2)favor it over alternatives
3)have judged the the disadvantages and are prepared for them
4)support it because they believe their party supports it
5)have faith in the political system
This isn't about the wording per se. Context matters.
To take a recent example: polls shows that Americans favor Obama over congressional republicans on economic issues. However, they still are (generally and marginally)opposed to his handling of economic issues and a strong minority of those who oppose the republicans do so out of a belief that they aren't cutting government enough. In other words, disillusioned republicans don't favor either party but aren't likely to support obama under any circumstances. You can look at the same polls with regard to china and democracy. Chinese both are satisfied with the direction of their country and want more democracy. If you aren't willing to consider context you can choose either poll to support your position.
It reminds me of a friend of mine who grew up around poor people and was aware of how they struggle with money. Her family wasn't in poverty, but they weren't flush with money either. At around age 30, she made her way into the finance industry and started pulling down six figures. She mentioned that she got a raise recently, but had been infected by the finance mindset that any raise is tainted by the marginal tax rate, and was bitching that 'half of it went to the government' (for values where .37 = half, it seems).
Anyway, her version of the story went 'the government taxes so much money out that I can't give as much to my single-mother sister who is scraping by on welfare - they should lower taxes [for this reason]'. I asked what about all the other single mothers our there that don't have sisters in high finance - reducing the tax yield would mean they get less. My friend changed her tune pretty fast and realised that hey, she was actually in a good position and should be positive about it instead of negative.
The thing is, my friend wasn't someone who grew up amongst wealth, she knew what it was like to be poor, and she herself had worked the same shitty jobs as poor people everywhere. She was smart, and knew deeply about social opportunity (she dated a fiery, forthright anarchist for many years). People just tend to have short memories and short forethought and tend to think that what concerns them in the present is the most important thing always.