The efficient market theory is that all information, public and private, is incorporated into all stock prices at all times, so there's no point to researching companies to try to outperform the stock market. I don't think anyone believes it is literally true, just that it is a good model for most stocks most of the time. If everyone believed it then no one would bother to research stocks, and the markets would be less efficient.
That's a far more limited claim. There's a very common claim that "markets are efficient". Extending to all markets all the time. Explaining why market based solutions are the best way to provide health care and education and other (arguably common) goods.
But many markets (the labor market, the health care market, etc) work in such a way that it's ridiculous to assume that most participants know all public and private information all or most of the time.
The EMH is specific to asset markets. So stocks and bonds basically. No (respectable, well-known) economist has ever argued that all markets are always efficient.
Also what people mean when they say markets are efficient in other contexts means something very different. The EMH specifically describes how quickly and what kinds of information get incorporated into asset prices. It doesn't make claims about how markets organize production or optimize utility without centralized direction, which is usually what people mean when they say free markets are efficient.
I think a lot of confusion has arisen from the very general-sounding name of the hypothesis which does not reflect its relatively narrow claims. Not that I buy the EMH (no pun intended), but that's a different story.
EMH that this paper refers to is about financial markets. In other areas of econ, no one is claiming that markets are efficient, in fact most of econ talks about "first best", "second best", "losses in efficiency", etc.
So called "Perfect Information", as you point out, is an important requirement of the efficient market hypothesis, but not the only one. It also requires fungible goods (e.g., if I start making bicycles that are better and cheaper than your bicycles, they will necessarily sell more. That's often not the case, especially if there is strong branding).
Another requirement is that entrepreneurs have fair market access to capital. This assumption is true in many developed countries, but not in many developing countries where you need connections to obtain a loan. But even in developed countries, it's difficult to get extremely large loans on the merits of the financials alone, which explains why large infrastructure projects like highways, airports, and power-plants often don't get built solely by the private sector.
Private information here refers to whatever insights you bring with your research. The point is that if you gain some private information (by being a quant, analyst, etc.) you would immediately trade on it and the price would then start reflecting your views.
Also EMH, and that's the point Fama uses a lot to battle critics, does not put a time frame on when the price adjustment happens.
