I agree with the assertion that the city manager is perhaps the best, if not the most democratic, strategy for running a government, in Texas or elsewhere. That said, however, they are usually not given long contracts, in fact sometimes shorter than the terms of elected officials, and are therefore as subject to politics as actual politicians are, sometimes even more so. Where city politicians are answerable to the electorate for their elected positions, city managers are answerable to city politicians for their livelihoods.
The fourth form of government, by the way, is the town meeting, which, even if the meetings are conducted among elected representatives, is best suited for very small communities. In Texas, I could only see it used in the more remote, rural regions of the western part of the state.
I think #2 makes an excellent point in terms of the city manager system being one of the best forms of government in Texas. One of the massive problems with politics is the way that popularity plays such a key role. It becomes very hard for candidates to make long-term decisions that may be unpopular in the short term but are actually the best for the country. Having somebody who is not elected therefore is a great benefit, as they do not need to worry about how the public feels about them in the same way that elected politicians do.
To me, there really are three systems with two sort of subsets of one of them. The three main ones are
- City manager-council
- City commission
Within the mayor-council system, you can have a strong mayor system and a weak mayor system. So maybe that's why your teacher/book talks about 4 forms.
As far as efficiency goes, I would say the city manager system is the best because you have an unelected technocrat running the city. That person can do what he/she thinks best without having to always worry about being reelected. The city council still has control so it's democratic, but the chief executive has more freedom.