What to write as 'purpose of journey' when visiting Estonia for a summer school during 4 days?

A summer school is clearly study (even though summer schools also have the social purpose of making international contacts in your research community); check that.

Also, don't sweat too much about the precise "correct" answer to that question. Other than "airport transit", there is no formal difference between Schengen visas based on which purpose you check; the question is just there to help the processing of the application and be an easy way for applicants in common situations to declare what their common situation is without having to phrase it themselves. The answer serves to clue the examiner in to which general kind of story your documentation will be telling, but as long as the totality of your application gives an honest and complete picture of your plans, they are not going to nitpick about which exact category of purpose they would have mapped those plans to.

I'll have to disagree with Henning. Most "summer schools" in computer sciences are not schools. They are not credit bearing, are short in duration, and consist of talks and workshops by active researchers in computer science on topics of research interest; attenders are also usually not required to be students (they could be professionals or even faculty themselves). Moreover, attending them don't enroll you in a program of academic study.

If they meet those criteria, they are a form of academic conference, not a program of study. You should therefore state that your purpose of visit is business/cultural, not study. If the summer school is credit-bearing or does constitute a program of academic study, though, then you will indeed be studying.