You can use it as a centralized repo, and it works quite well. In fact, it's how I'm doing development with LibreOffice via gerrit. However, even there I find it easier. I find my workflow is to work on a branch, make sure I have clean commits (I use rebase pretty frequently) and branch and merge often. Frankly, it's easier to work with. I find the staging model to be really excellent, and find it just easier to move up and down commits.
But looking at the quotes:
Git seems to be the "new, shiny, cool" thing.
No, no it's not.
Two modes of creating repositories
Well, sure. It's literally just a shortcut for setting up groups and then changing a global setting. Repos can go from bare to shared reasonably easily. I fail to see the point.
checkout vs. clone
Checkout checks out a branch of path to the working directory. Clone gets an entire repository and "clones" it to the current directory. There is no "versus" here, they are two completely different operations.
commit vs. push
Commit records the changes to the local repository. Push basically takes a ref (a mapping to a sha-1 value that identifies a commit in the tree) and updates a ref in a remote repository. In essence, you are pushing your changes to the remote repository. Again, the thinking of a VCS person, not someone used to DVCSes.
You have to know which commands work locally and which work with "the server"
Only if you think in terms of centralized repositories. A DVCS is distributed. It's a completely different way of thinking. I find it superior in very many ways.
Git has the advantage that it's MUCH better suited if some developers are not always connected to the master repository.
The whole point of git is that you have your version, and you either pull from or push to different repositories and and when needed. The whole idea of centralization is not core to it at all.