externalities are not technologically but rather socially determined. There are no inherent properties of a good or service producing external effects, therefore, citizens have to use the political process to determine what is to be considered to be an externality.
I tend to use "public good" and "positive externality" interchangeably. The idea is that Sally and Tom benefit from the actions of Joe, but Sally and Tom do not compensate Joe for those benefits. A wise and omniscient government needs to step in and tax Sally and Tom to subsidize Joe's actions.
If I provide a private security guard to patrol the neighborhood, the neighbors who don't pay for the security guard nonetheless cannot be excluded from enjoying some of the benefits. Moreover, the benefits that they get from the security guard do not detract from the benefits I get (i.e., our consumption of security services is "non-rivalrous"). Non-excludability and non-rivalry are thought to be conditions of public goods.
Even though the security guard is not excludable and non-rivalrous, it may not be a public good. Perhaps I get enough benefits from the security guard that I am willing to pay for it, without needing contributions from the neighbors. In that case, the positive externality is not big enough to worry about. Conversely, the total value of the security guard to everyone may be less than the cost of providing it. In that case, we should not collect taxes and pay for the security guard.
For education, the positive externality is the benefits that accrue to me from your education. I think that those benefits tend to be pretty small. You get a higher income, and most of those benefits flow to you. I get some of the benefits, because you are more likely to pay taxes and less likely to require government transfers, so that my tax obligations can be correspondingly reduced.
You also get the consumption benefits of your education. I personally don't benefit from your experiments with drugs, sex, rock'n'roll. Nor do I particularly care that you take a class in art appreciation or get tickets to your school's basketball games.
Finally, you are supposed to be a better citizen because of education, and I should be happy about that. But if what you learn is that profits are evil, man-made global warming is beyond doubt, and it is wrong to question that gender differences are socially constructed, then from my perspective your education is not making you a better citizen.
If the higher income that you get from education is due to its signaling effects, then that is a classic negative externality. The investment in the signal is wasteful, and your investment forces others to make a wasteful investment.
On the whole, the case for taxing education rather than subsidizing it is really quite plausible. It is counter-intuitive, perhaps, but the case for free trade is also counter-intuitive to most people.
The Frey-Eichenberger quote is from p. 30 of The New Democratic Federalism for Europe, where they lay out the idea of FOCJ, which stands for functional, overlapping and competing jurisdictions. It's a form of competitive government. So far (I'm only about a third through the book), they seem to avoid solving the three tough issues with competitive government in general and their idea in particular. These questions are:
1. How should income redistribution be handled?
2. How should paternalism be handled?
3. What should be the process for determining what is or is not a public good?
The quote above pertains to the third problem.
For the first problem, if you have many government agencies pursuing income redistribution, you could get messy results. For example, in the U.S., there are so many programs that have income cutoffs that the marginal tax rate on poor people is alarmingly high.
The second problem, paternalism, crops up because with FOCJ people are supposed to be able to opt out of "one-size-fits-all" programs like Social Security. As long as you're at least a little paternalistic, you worry about what they will opt into. But which government agency is in charge of determining how that paternalism applies?