The Preface states: "The converse of this is that democracy has prerequisites for its participants. The system is government by the people, but for it to function properly the people must be well-educated."

Does "government by the people" mean (or can it be translated as) "power by the people", "ruling regime by the people,” “elected by the people,” etc? Also, does "must be well-educated" mean that the people must be literate? Or they must have a degree from an educational establishment?

The statement in the Preface can be expanded as follows:

"The system is government by the people (the public, ultimately, is responsible for government decision-making, even if elected officials make the actual day-to-day decisions), but for it to function properly the people must be well educated (they must work to understand the larger issues that have an impact on their lives)."

As elaboration, democracy is self-government, but under the representative system we elect officials to do the job. The key issue is who makes the actual decisions. Again, in the representative system the officials make the decisions, but if we don't like their choices we can remove them from power and install someone else. We retain this power, and responsibility.
I wouldn't use the phrase "popular regime," or the "people's regime," etc., to describe democracy. "Regime" is a word that is associated with authoritarian rule.
For education, literacy is a requirement, but not a university degree. Of more importance is desire. You can be well-educated even if you have no formal education, if you study on your own and with the help of others: if you want to learn and work at it until you do.
The requirements for the electorate are described more extensively in the lesson: The people in a democracy.