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Question: Ayn Rand says man's noblest activity is "productive achievement." What is productive achievement? Someone said: "the application of reason to the problem of existence."
1. Linus Torvalds creates Linux and gives it to the world for free. Is this productive achievement even though he doesn't make any money out of it?
2. Gary Kasparov (World Chess Champion 1985-2000) makes a career out of the game of chess. Is devotion of one's life to a mere game productive achievement?
3. A certain product (I have several specific products in mind, but let's not get lost in discussion of concrete particulars) is poison. The manufacturer KNOWS it is poison, but sells it as food. The customers DON'T know it's poison and would not buy it if they did know, and the manufacturer knows this and therefore is careful to not tell them the truth about his product. Is this productive achievement?
Answer: The definition you quote for productive achievement accurately states the Objectivist view (though Ayn Rand's wording is slightly different; her statement is "production is the application of reason to the problem of survival" [Capitalism: The Unknown Ideal, p. 17]). Productive achievement is the use of reason to create the values man needs, directly or indirectly, for his survival. Productive achievement would be possible to someone living on a desert island (in fact it would be the only way he could possibly survive), but, in a complex society based on division of labor, productive achievement will almost always involve trade, and therefore require the creation of value for other people.
Using this basic understanding of productive achievement to address your examples: on example 1, there is nothing in the Objectivist view of productive achievement that requires that one make money from it. Certainly Torvalds has created an important value that is useful to a lot of people, and his is a great and admirable achievement. However, it is important to note that while declining to make money from his creation doesn't detract from Torvalds's achievement, it also doesn't add to it; Objectivism completely rejects the conventional view that a creator who makes his creation available for free is in any way greater or more noble than a creator who turns his creation into a profitable business.
On example 3, there are two points to be made. First, productive achievement does require the creation of a genuine value; fraud, or catering to people's irrationality, are not cases of productive achievement. To the extent that someone obtains money by selling something that is of no genuine value, and that people are willing to buy only out of ignorance or to satisfy an irrational need, he is not productive.
Second, it is important to understand that there is a vast range of legitimate options in how people choose to live their lives. Referring to your example of food, there are many different values one gets from food that directly or indirectly affect one's survival, and therefore many different considerations in choosing what to eat: the long-term healthfulness of the food, its price, the amount of time and effort needed to prepare it, and the physical enjoyment you get from it; different people will make different trade-offs on these considerations. Many people assume that their choice of trade-offs between these considerations is the only valid one. You often hear the claim that the fact that many people eat a certain food can only be the result of their ignorance, and accusations against the manufacturer of deliberately concealing the truth; but generally such claims have no basis, and simply result from a refusal to accept that people can make different choices. It is possible to disapprove of a product, have good rational reasons for declining to buy it, and still recognize that creating that product for those people who do choose to use it is a productive achievement.
Your example 2 is the hardest to answer, and the one that is least addressed in the existing Objectivist literature. I will give you my view (which I think is consistent with Rand's discussion in her article "An Open Letter to Boris Spassky," reprinted in Philosophy: Who Needs It), but I know many intelligent and knowledgeable Objectivists who will disagree with me. In playing a game such as chess, one is applying reason, not to any aspect of the problem of survival, and not to transforming reality in any way, but to artificial problems of manipulating an artificial set of concretes by arbitrary rules not connected to anything outside of the game. Playing a game therefore cannot qualify as productive achievement. It is a legitimate leisure activity, for relaxation and for honing one's skills; but it is not productive, and is not a legitimate career to which to devote one's life.