What are the minimum contributions required for co-authorship
Short answer: free authorship, also known as gift authorship, is a clear violation of research and publication ethics. The limit between “small contribution” and “no contribution” is not, however, easily defined; different fields put it at different levels. 



I don't think the answer actually states that. It lists a series of contributions that are, in some field, considered important enough to warrant (in some combinations) authorship on resulting papers:


  they get grant money, they train you to use the lab, they train you to do statistics... or they might make suggestions for the research design, the main theoretical focus of the presentation / manuscript


All journals (or publishers) have policies or guidelines on how authorship should be determined. In all cases, it involves significant scientific or technical contributions to the work published. Authorship determination has to be weighted in each individual case, as no two situations are identical (and not simple rule of thumb can encompass all possible situations, as Peter Jansson highlights).

There are so many ways in which people can contribute to an intellectual work, such as a research project and academic papers. It happens that different fields of research have different habits in authorship determination, giving more or less weight to different types of contributions. My own background is in physics and chemistry, where authorship tend to be more generous that, say, computer science or mathematics. I'll thus argue two examples of what you (and some fields) may consider dubious basis for authorship, but which in my field would be considered fairly standard:


Getting funding. In this age, getting funding most often requires writing a grant proposal for a specific research program, with good and novel ideas, and convincing a tough crowd of other scientists (in a competitive environment) that your program is a good use of taxpayers' money. Thus, in most cases, the person who provides the funding also provides a clear scientific contribution: they identified an important problem to be solved, and provided a general framework for solving it. That's an important part of research! Identifying the right question to ask yourself is half of the job, it is known. (Yeah, I'm trying my hand at Dothraki style in academic context. By the way, thank you for reading so far down my answer.)
Supervising. Whether or not the supervisor actually provided the student with the research project in the first place, supervision implies guidance of the student, which is definitely a scientific contribution. The supervisor will, in many cases, provide a broader view of the field and ideas for related problems relevant to the research, scientific background, and advice on how to use one's research time most efficiently. All of that is highly valuable, and contributes to the publication.

(I've read somewhere “but the supervisor is paid for this, it's his job, so he shouldn't be awarded authorship” — that's plain stupid, pretty much everyone gets paid to do research, thus by that argument most papers would be authorless.)


In both cases, funding and supervising imply scientific contributions, which are the reason for authorship.